Move-specific suggestions:
• Minimize
Confers a buff (volatile status) that caps the Base Power of any move used against this pokémon at 60 BP. Doesn't stack. Stomp and Steamroller ignore this cap.
Motivation: This is one step toward scrapping evasion clause. Minimize has a fairly restricted distribution, and its previous effect was fairly powerful, indirectly boosting both sides (physical and special) of a pokémon's defensive potential. This is retained in the new effect, which especially acts as a counter-measure against overused high BP moves with downsides like Draco Meteor. The new guaranteed crit moves moves and the buffed multi-hit moves act as checks against the new minimize.
Thematic justification: If you imagine an attack exerting a certain "pressure" against the opponent, shrinking the target reduces the amount of force that can be effectively delivered against it. If you halve the size of a sail, the wind can only "attack" it half as hard.
• OHKO moves
Fissure → Ground, physical
Sheer Cold → Ice, special
Guillotine → Dark, physical
Horn Drill → Steel, physical
All four: These moves always miss the first time they are used, and always fail against sturdy pokémon. They have 100BP and 90% accuracy with the additional effect that a successful hit will OHKO any pokémon weak to them.
Motivation: Lets you scrap OHKO clause. The revised moves are not nearly as luck dependent but still retain some of their former flavour and wall-breaking potential. The fact that they always miss the first time warns any opponent who might be vulnerable to the one-hit knock-out effect. Changing the types of guillotine and horn drill provides some coverage to the pokémon that learn these moves, but I think their power level is still not very high.
• Double Team
Just brainstorming on this one; I have a few ideas for it that are perhaps too far out in left-field. Maybe it could set up an effect on your side of the field (limited duration) so that whenever your pokémon switches out, the incoming pokémon inherits its resistances and immunities, or even gets STAB on its move types. That would be cool but a little confusing. You wouldn't want to make it too powerful, because almost everything gets Double Team.
• Power Whip
Maybe it should get an accuracy nerf? It's a physical grass move that's as powerful as, and more accurate than, hydro pump.
• Hex
When you buffed shadow ball, this move's niche got demolished. I'm not sure the best way to strengthen it again.
• Solar Beam
You added effects that happen on the charge turn of some 2 turn moves, but then what happens to Solar Beam under sun? If it skips the healing you would get on the charge turn, that can create an odd corner case. If your pokémon is bulky and durable, you might be put in the odd position of thinking "I sure wish it weren't SUNNY as I'm using this SOLAR beam attack." I suggest that under sun all the effects of the move should be compressed into one turn so you get the healing too. I'd reduce the healing to 1 / 3 Max HP so that it's not overpowered.
• Dream Eater
Has a 70% chance to add 1 to a sleeping target's sleep counter. Hits a sleeping target that switches out pursuit-like, but not with doubled power. Signature move for hypno?
• Mimic
Works like sketch in-battle. That is, once you've copied an opponent's move, you retain it even if you switch out.
• Gastro Acid
Parallel with Acid Spray, it gives the target -2 SpDef in addition to its ability-negating effect.
• Natural Gift
This is an awfully complicated attack for it to be as useless as it is. Talk about wasted complexity... +40 BP across the board? That might result in a LOL-worthy mixed-attack harvest exeggutor set.
• Pay Day
Half-joking here, but a 60 BP attack that doubles to 120 BP and heals 1/3 Max HP if the turn number is divisible by 7.
• Spit up
Hits with 200 BP and removes only 1 stockpile boost.
• Swallow
Heals 2/3 Max HP and removes 1 stockpile boost.