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Firstly, stat boosts don’t stack. Moves like swords dance and nasty plot give the same attack boost as calm mind and bulk up. However, calm mind and bulk up wear off faster than swords dance and nasty plot, since they boost defense and offense. Strong style increases how long the boost will last for, agile style decreases how long the boost will last for. For a more specific explanation, refer to the above post by Germaniac.

I’ve done a bunch of research, I haven’t found the exact way turn order is calculated, but from some playtesting I did discover a interesting detail. It seems that agile style moves actually lower the target’s turn order instead of raising yours. In a 3v1 against the same Pokémon all at the same level, using an agile move only changed the targeted pokemon’s turn. IE
Targeting Paras 1 Moved it’s turn after user’s next turn
Paras 2 and 3’s turns remained the same, still occurring before my next turn
Changing the target to Paras 2 moves it’s turn after my second turn
Paras 1 and 3 still move before my second turn

I haven’t done super thorough testing, so this could be caused by something else for all I know, and I haven’t found any formulas on calculating turn order to confirm or deny this yet. I’ll make any changes as I discover anything.

Very interesting, that'd explain why most moves have the same ActionSpeedMod and AgileActionSpeedMod. I think the most likely modifier for the opponent's delay is, despite the name, StrongTargetActionSpeedMod. They're only 0 for moves that target self, Mimic, and Struggle, and the latter 2 are unable to be used in the styles.

On another note, I just checked that some moves with 0 flinch rate has non-0 TargetActionSpeedMod and vice versa, so I guess I'm back to square one on this.

Over these next few days I plan to try and crack the damage calculation formula, if someone hasn’t beaten me to it. I plan on using three subjects: Manaphy, Phione, and Arceus, since they all have equal HP, Def, and SpD (80,100, 120), all with max EL and natures that neither increase nor decrease their stats, and all at level 100. I’ll use each of them against the NPC trainers in jubilife village and record the damage taken from each moves used several times. Then after getting enough data, I’ll try to find the correlation. If there’s anything in this method I’m missing, let me know!

I was trying to catch an alpha and accidentally burned it at low HP, but it looked like it had more than 1/16th of its HP left so I thought it was fine.
It wasn't.

Max HP

Damage per turn from burn

235

20

337

29

Max HP

Damage per turn from frostbite

235

20

272

23

337

29

Both burn and frostbite deal 1/12 max HP, rounded up!

Firstly, stat boosts don’t stack. Moves like swords dance and nasty plot give the same attack boost as calm mind and bulk up. However, calm mind and bulk up wear off faster than swords dance and nasty plot, since they boost defense and offense. Strong style increases how long the boost will last for, agile style decreases how long the boost will last for. For a more specific explanation, refer to the above post by Germaniac.

To add on this, what about Double Hit? It's the only boosting move that specifies its effect, but it may affect base power instead of stats? And I guess it can stack with standard boost moves for this reason (also, how much do Swords Dance/Calm Mind/Bulk Up/Nasty Plot boost your stats? x1.5?)

I continued logging drowsy turns as I played through, culminating in 112/333 drowsy/total turns (this includes my initial 100 trials from the previous page). So I'm fairly certain drowsy is indeed 1/3 chance to not move. Given the similarity drowsiness has to paralysis, it might be worth checking if the paralysis rate is now also 1/3 instead of its usual 1/4.

To add on this, what about Double Hit? It's the only boosting move that specifies its effect, but it may affect base power instead of stats? And I guess it can stack with standard boost moves for this reason (also, how much do Swords Dance/Calm Mind/Bulk Up/Nasty Plot boost your stats? x1.5?)

I’m still in the process of gathering sample data for cracking the damage formula, but here’s what I’ve gotten so far. I haven’t been able to start working on the formula because I haven’t been able to find out the stats of the NPC’s Mothim. However, here are the values I have for now if anyone wants to experiment around:
(75 Base Power) Air Slash Damage:
75
77
80
72

(80 Base power) Bug Buzz Damage:
75
84
77
76

Stats of level 52 Serious Manaphy:
193 Def
193 SpS
281 HP

Opponent was level 55 Mothim used by Wenton in Ingo’s battles. Now determined its SpA is 154.

From a bit of messing around, if Wenton’s Mothim is fully ELed, and has a SpA raising nature, the damage formula is the same as mainline. I doubt this is the case, but that can’t really be determined until I find the stats for the Mothim.

With the stat calculations more or less known, I decided to investigate a little into his Alakazam's stats. OHKOing it with Double-Edge inflicted 61 damage normally, and 93 damage from a strong version (which increases recoil to 50% of damage dealt). This roughly corresponds to a 0 EL Alakazam which would have 185 or so HP with the formula as it's known so far (not knowing when or if it rounds or floors etc will skew the results slightly). I then took the damage range versus my Wyrdeer and used a damage calculator for gen 8 to see what special attack it corresponded to. I found it actually almost exactly corresponded to an EL10 +SpA Alakazam, 373 SpA (374 from formula). This seemed pretty on the nose, but it would only be relevant if the damage formula was the same, and that is, to say the least, a big if. Getting some numbers from its Psychic on my Avalugg almost immediately showed that the formula had indeed changed in some way.

Pokemon typically uses a linear damage calculation; if you double the defense you'll halve the damage (more or less). Or in this more particular case, if Wyrdeer has ~1.696 more special defense and takes a move that's 2/3 as strong, it'll take around 39% of the damage, comparatively. Yet, Wyrdeer is taking noticeably more than that - it's taking over half of what Avalugg's taking. I don't really have an entirely satisfying conclusion for this unfortunately, but I am at least fairly sure there is some kind of dampening effect on your defense stat in this game. Or more simply, increasing your defense stat doesn't matter as much anymore.

I took some numbers versus my other team members and they mostly showed the same result - compared to the lower special defense Pokemon, those with higher special defense were taking more damage (relatively) than they would with the typical damage formula. I did notice one weird exception comparing my Wyrdeer and Sylveon, though. My Sylveon has 293 special defense, which compared to Wyrdeer's 190 yet also taking 50% more from the move power (STAB Psychic = 80x1.5 compared to unSTAB Tri Attack at 80) would mean they would take almost the same damage with the typical formula.

The weird part was that if the effect of defense was reduced, Sylveon should've been the one more hurt by that (as it has the higher defense) and start to take more damage. Yet, it actually took less damage than Wyrdeer. Comparing all my other pairs, I realized one crucial difference - Wyrdeer was the only Pokemon not taking STAB Psychic. I casually mentioned that STAB provided a 50% boost to the BP, but that's just an assumption based on previous games, the game never actually states this. If the damage multiplier from STAB is less than that, then it makes sense for Sylveon to take less damage, and it wouldn't matter for most of the other comparisons since I'd be comparing Psychic to Psychic. It would skew my Wyrdeer vs Avalugg comparison, but Avalugg takes a comically high amount of extra damage anyways so unless STAB straight up didn't exist the same conclusion would hold. Anyways, Ingo's Alakazam also knows Shadow Ball and Tri Attack, which if used on Sylveon (after stalling out Psychic) provides a great way to check for STAB since they're the same BP as Psychic:

Shadow Ball/Tri Attack vs Level 78 EL6 293 sdef Sylveon

Indeed - STAB does not appear to be a 1.5x multiplier in this game! It seems to be around 1.25x. That removes the earlier discrepancy, but I'm still not exactly sure how defense is being reduced in the damage formula. I'll continue to look into it, but for now here's the rest of my data:

Tri Attack on Wyrdeer with a defense boost

55, 56, 53, 53, 51, 57, 51, 58

This was a quick test to see if defense boost reduced damage by 1/3 as a +1 boost usually would. It seems like it does, though it's probably a damage reduction rather than an actual modifier to the defense, given the likely non-linear nature of defense in the damage formula.

Psychic on Sylveon with a defense drop

110, 100, 107, 111, 104, 114

Similar to the previous test with an entirely analogous conclusion.

With the stat calculations more or less known, I decided to investigate a little into his Alakazam's stats. OHKOing it with Double-Edge inflicted 61 damage normally, and 93 damage from a strong version (which increases recoil to 50% of damage dealt). This roughly corresponds to a 0 EL Alakazam which would have 185 or so HP with the formula as it's known so far (not knowing when or if it rounds or floors etc will skew the results slightly). I then took the damage range versus my Wyrdeer and used a damage calculator for gen 8 to see what special attack it corresponded to. I found it actually almost exactly corresponded to an EL10 +SpA Alakazam, 373 SpA (374 from formula). This seemed pretty on the nose, but it would only be relevant if the damage formula was the same, and that is, to say the least, a big if. Getting some numbers from its Psychic on my Avalugg almost immediately showed that the formula had indeed changed in some way.

Pokemon typically uses a linear damage calculation; if you double the defense you'll halve the damage (more or less). Or in this more particular case, if Wyrdeer has ~1.696 more special defense and takes a move that's 2/3 as strong, it'll take around 39% of the damage, comparatively. Yet, Wyrdeer is taking noticeably more than that - it's taking over half of what Avalugg's taking. I don't really have an entirely satisfying conclusion for this unfortunately, but I am at least fairly sure there is some kind of dampening effect on your defense stat in this game. Or more simply, increasing your defense stat doesn't matter as much anymore.

I took some numbers versus my other team members and they mostly showed the same result - compared to the lower special defense Pokemon, those with higher special defense were taking more damage (relatively) than they would with the typical damage formula. I did notice one weird exception comparing my Wyrdeer and Sylveon, though. My Sylveon has 293 special defense, which compared to Wyrdeer's 190 yet also taking 50% more from the move power (STAB Psychic = 80x1.5 compared to unSTAB Tri Attack at 80) would mean they would take almost the same damage with the typical formula.

The weird part was that if the effect of defense was reduced, Sylveon should've been the one more hurt by that (as it has the higher defense) and start to take more damage. Yet, it actually took less damage than Wyrdeer. Comparing all my other pairs, I realized one crucial difference - Wyrdeer was the only Pokemon not taking STAB Psychic. I casually mentioned that STAB provided a 50% boost to the BP, but that's just an assumption based on previous games, the game never actually states this. If the damage multiplier from STAB is less than that, then it makes sense for Sylveon to take less damage, and it wouldn't matter for most of the other comparisons since I'd be comparing Psychic to Psychic. It would skew my Wyrdeer vs Avalugg comparison, but Avalugg takes a comically high amount of extra damage anyways so unless STAB straight up didn't exist the same conclusion would hold. Anyways, Ingo's Alakazam also knows Shadow Ball and Tri Attack, which if used on Sylveon (after stalling out Psychic) provides a great way to check for STAB since they're the same BP as Psychic:

Shadow Ball/Tri Attack vs Level 78 EL6 293 sdef Sylveon

Indeed - STAB does not appear to be a 1.5x multiplier in this game! It seems to be around 1.25x. That removes the earlier discrepancy, but I'm still not exactly sure how defense is being reduced in the damage formula. I'll continue to look into it, but for now here's the rest of my data:

Tri Attack on Wyrdeer with a defense boost

55, 56, 53, 53, 51, 57, 51, 58

This was a quick test to see if defense boost reduced damage by 1/3 as a +1 boost usually would. It seems like it does, though it's probably a damage reduction rather than an actual modifier to the defense, given the likely non-linear nature of defense in the damage formula.

Psychic on Sylveon with a defense drop

110, 100, 107, 111, 104, 114

Similar to the previous test with an entirely analogous conclusion.

This definitely debunks any possibility that the damage formula was the same. Thought there was a small chance this would be easy. But on an unrelated note, perhaps stab and damage are connected to a Pokemon’s level? Not just in the sense that stats improve with level. They’ve certainly done that with the new EL and stat calculations, using levels a lot more in the formulas. Just a hunch though. It is possible that the damage formula isn’t the same but is instead slightly modified. I think for testing purposes starting with gen mainline damage formula won’t be a horrible idea. The values aren’t crazy off after all, at least from the minimal testing I’ve been able to do (I severely underestimated how long it would take to level up three mythicals to level 100)

Fixated - how much more damage does a fixated move deal, and how much more damage does the user take from attacks during the effect?

Splinters - what is the formula for calculating its damage? It's not a consistent fraction of max HP. There's a 25 in the move data for each of the moves, perhaps a power value?

Obscured - what is the boost to evasiveness?

Drowsy - chance of being too drowsy to move? How much more damage does a drowsy Pokemon take from attacks?

I am under the impression, based on using lots of Splinter moves, that the type of splinter MAY affect the damage, similarly to Stealth Rock's damage was affected by the type chart. I'm also under the impression that level disparity may also play a role in the damage done at the end of the turn. I say these things because it appears different Splinter moves do more damage to different type combinations, but this may also be level disparity. I haven't properly tested these mechanics, only sharing my experience with them. I used Stealth Rock on level 62 Alpha Blissey from my 40 something Empoleon and the splinters barely did any damage. But the splinters did ~25% to a Flying type that was 7 levels lower level than my Pokemon, so...

Serious has been used before as the standard neutral nature for NPCs. Still doesn't explain why exactly, but it is consistent with previous trainer data.

After several different tests I think I have a candidate for the damage formula. I normally wouldn't post it on here, but I don't currently have the resources in game to keep testing it, so if anyone else wants to keep testing it and prove/disprove while I farm up more resources, be my guest! No hard feelings if I'm completely wrong, and any help is greatly appreciated! The formula that has worked with the little bit of testing I did is:
((attackerlevel/50)(AttackPower)(AttackerAttackStat))/(defenderdefense))(stab)(x)

Based on what I read above, I used stab as 1.25, and X is a RNG value between .85 and 1. Both attack and defense stats are the actual stats, not the base stats. With my Manaphy with 193 defense, the formula accurately predicted the damage it took from a level 55 no EL or nature Mothim, both from Bug Buzz and Air Slash. However I ran out of potions and potion materials, so I had to cut the testing short before I could try the next mon.

Forgot to mention it rounds up
Here’s a better visual:

After several different tests I think I have a candidate for the damage formula. I normally wouldn't post it on here, but I don't currently have the resources in game to keep testing it, so if anyone else wants to keep testing it and prove/disprove while I farm up more resources, be my guest! No hard feelings if I'm completely wrong, and any help is greatly appreciated! The formula that has worked with the little bit of testing I did is:
((attackerlevel/50)(AttackPower)(AttackerAttackStat))/(defenderdefense))(stab)(x)

Based on what I read above, I used stab as 1.25, and X is a RNG value between .85 and 1. Both attack and defense stats are the actual stats, not the base stats. With my Manaphy with 193 defense, the formula accurately predicted the damage it took from a level 55 no EL or nature Mothim, both from Bug Buzz and Air Slash. However I ran out of potions and potion materials, so I had to cut the testing short before I could try the next mon.

Factoring in STAB only being 1.25x, a simple comparative analysis indicates that Stone Edge only being ~1.65x stronger than STAB Aerial Ace puts it around 250 BP, so I think "4x" effective moves now hit for 2.5x damage.

Now for the more complicated part. To continue from my previous post about the effect of defense, I was playing around with various dampening effects and found that the one that made the various defense/BP ratios stay consistent with the ratio of my observed damages on their respective targets was an additive effect of plus 50. So my assumption is that defense is replaced with defense+50 in the formula.

Attack is where things get kind of insane, though. Let's look at this pair of observed damage ranges:

Level 67 Probopass Earth Power vs 190 sdef Wyrdeer

82, 71, 80, 74, 80, 80, 73, 71, 73, 77

Okay. So let's review: same level, same BP, same type effectiveness, same special defense. The only difference between these two ranges are the attack stats of the attackers. From Anubis's post, we know that Alakazam has 237 special attack, while Probopass has 144. Alakazam has 64% more special attack, so under the typical formula where the (special) attack is more or less linear, that would translate into a 64% increase in damage. Here, though? Uh...5-8% more (depending on which end you compare - the Probopass range is likely somewhat incomplete). This means the attack stat is severely dampened in the damage formula.

My first instinct was logarithms, and similar to what I did with defense above I compared attack/BP ratios with my the ratios of observed damage, this time from different attackers on the same target. Logarithms made them fairly inline with each other, though unfortunately dividing two logarithms of the same base always has the same result (log(numerator) with base denominator), so which base to use was a guessing game. I decided on an arbitrary base of 2, with an also arbitrary multiplier of x60 to make the damages line-up with what I had been seeing (using defense+50 in place of defense). The results mostly lined up with my observed within a few points, but while I had faced one lower leveled Pokemon (Level 55 Mothim from Wen), I decided I should try and get something from a much lower leveled Pokemon. I found a Level 27 Staravia, and had it attack my Yanmega until it mostly ran out of PP:

88, 100, 94, 98, 94, 90 (noticeably incomplete because this dumbass staravia missed 4/10 air slashes!!)

Catching it revealed it had 95 attack and 52 special attack. My formula produced fairly accurate values for Quick Attack and Aerial Ace, but only predicted a max value of 78 for Air Slash. This really drives the point of attack stats not mattering much though. Look how much more Air Slash does than Aerial Ace. Sure, Air Slash is stronger than Aerial Ace and Yanmega's defense is higher than its special defense, but in the traditional system Aerial Ace should still be about 10% stronger due to the much higher attack stat. Here, Air Slash instead is demonstrably around 60% stronger.

Anyways, I'll be honest and say I'm not really sure at all what the precise formula is. I've tinkered around a good bit with various combinations logarithm bases, additive effects, and multipliers, but while many are "close" they're consistently a little low here, a little high elsewhere, etc. It may not even involve a logarithm specifically. Regardless though, unless I'm overlooking something huge (maybe level of the defender does something now? that's the only real constant factor now) it really does seem like your attack stat matters fairly little in this game.

Data dump for those that want to mess around:

Level 68 Gliscor X-Scissor vs 179 def Electrode-H:

154, 170, 156, 162, 154, 162, 152, 154, 176, 180

Level 55 Mothim Air Slash vs 152 sdef Yanmega:

170, 170, 198, 194

Level 55 Mothim Bug Buzz vs 112 sdef Avalugg-H:

116, 121, 117, 117, 121, 115, 114, 133

Level 55 Mothim Bug Buzz vs 190 sdef Wyrdeer:

152, 170, 170, 172, 150, 160, 174

Level 67 Probopass Flash Cannon vs 293 sdef Sylveon:

144, 140, 130, 140, 122

Level 67 Probopass Flash Cannon/Power Gem vs 190 sdef Wyrdeer:

Factoring in STAB only being 1.25x, a simple comparative analysis indicates that Stone Edge only being ~1.65x stronger than STAB Aerial Ace puts it around 250 BP, so I think "4x" effective moves now hit for 2.5x damage.

Now for the more complicated part. To continue from my previous post about the effect of defense, I was playing around with various dampening effects and found that the one that made the various defense/BP ratios stay consistent with the ratio of my observed damages on their respective targets was an additive effect of plus 50. So my assumption is that defense is replaced with defense+50 in the formula.

Attack is where things get kind of insane, though. Let's look at this pair of observed damage ranges:

Level 67 Probopass Earth Power vs 190 sdef Wyrdeer

82, 71, 80, 74, 80, 80, 73, 71, 73, 77

Okay. So let's review: same level, same BP, same type effectiveness, same special defense. The only difference between these two ranges are the attack stats of the attackers. From Anubis's post, we know that Alakazam has 237 special attack, while Probopass has 144. Alakazam has 64% more special attack, so under the typical formula where the (special) attack is more or less linear, that would translate into a 64% increase in damage. Here, though? Uh...5-8% more (depending on which end you compare - the Probopass range is likely somewhat incomplete). This means the attack stat is severely dampened in the damage formula.

My first instinct was logarithms, and similar to what I did with defense above I compared attack/BP ratios with my the ratios of observed damage, this time from different attackers on the same target. Logarithms made them fairly inline with each other, though unfortunately dividing two logarithms of the same base always has the same result (log(numerator) with base denominator), so which base to use was a guessing game. I decided on an arbitrary base of 2, with an also arbitrary multiplier of x60 to make the damages line-up with what I had been seeing (using defense+50 in place of defense). The results mostly lined up with my observed within a few points, but while I had faced one lower leveled Pokemon (Level 55 Mothim from Wen), I decided I should try and get something from a much lower leveled Pokemon. I found a Level 27 Staravia, and had it attack my Yanmega until it mostly ran out of PP:

88, 100, 94, 98, 94, 90 (noticeably incomplete because this dumbass staravia missed 4/10 air slashes!!)

Catching it revealed it had 95 attack and 52 special attack. My formula produced fairly accurate values for Quick Attack and Aerial Ace, but only predicted a max value of 78 for Air Slash. This really drives the point of attack stats not mattering much though. Look how much more Air Slash does than Aerial Ace. Sure, Air Slash is stronger than Aerial Ace and Yanmega's defense is higher than its special defense, but in the traditional system Aerial Ace should still be about 10% stronger due to the much higher attack stat. Here, Air Slash instead is demonstrably around 60% stronger.

Anyways, I'll be honest and say I'm not really sure at all what the precise formula is. I've tinkered around a good bit with various combinations logarithm bases, additive effects, and multipliers, but while many are "close" they're consistently a little low here, a little high elsewhere, etc. It may not even involve a logarithm specifically. Regardless though, unless I'm overlooking something huge (maybe level of the defender does something now? that's the only real constant factor now) it really does seem like your attack stat matters fairly little in this game.

Data dump for those that want to mess around:

Level 68 Gliscor X-Scissor vs 179 def Electrode-H:

154, 170, 156, 162, 154, 162, 152, 154, 176, 180

Level 55 Mothim Air Slash vs 152 sdef Yanmega:

170, 170, 198, 194

Level 55 Mothim Bug Buzz vs 112 sdef Avalugg-H:

116, 121, 117, 117, 121, 115, 114, 133

Level 55 Mothim Bug Buzz vs 190 sdef Wyrdeer:

152, 170, 170, 172, 150, 160, 174

Level 67 Probopass Flash Cannon vs 293 sdef Sylveon:

144, 140, 130, 140, 122

Level 67 Probopass Flash Cannon/Power Gem vs 190 sdef Wyrdeer:

How does Self-Destruct recoil work now? Is the recoil fixed to X HP lost, no matter the foes? From using it with Azelf, I've seen that it doesn't K.O yourself anymore.

What is the power boost from the rain and the drought for, respectively, Water- and Fire-types attacks?

Also, can't wait to see how research for the damage formula turns out!

How does Self-Destruct recoil work now? Is the recoil fixed to X HP lost, no matter the foes? From using it with Azelf, I've seen that it doesn't K.O yourself anymore.

This is actually part of the move data linked in the OP! Self-Destruct removes 80% of the user's max HP normally, 90% for strong style, and 70% for agile style.

I think so. If you compare the damage between Staravia's Quick Attack and Aerial Ace, it's roughly triple the value (x1.5 BP difference, x2 type effectiveness). Technically it's a little lower (~1.9x factoring out BP difference), but it makes more sense to chalk that up to slightly incomplete ranges (my tests are not exhaustive, I just stop when it looks like I've got the extremes of the range) and/or flooring weirdness.