Shadow Tag.
In most cases Levitate, unless the pokemon in question has a better ability to warrant the use Magnet Rise (Why bother using Magnezone if it didn't have Magnet Pull?).
Imposter, unless you're playing doubles (I remember running an uber team with Transform Mew and Scarf Kyogre that resulted in double STAB Water Spout spam. Good times.)
Speed boost, although Agility does give you more speed faster.
Purely from a singles standpoint, No Guard, although it is possible Telekinesis may have some other uses in doubles for helping pokemon with ground weaks on teams that like to spam EQ, or if you don't want to run Gravity but want to use moves with less acc, etc etc (although I guess this is getting away from what you're trying to prove).
Scrappy
Magic Bounce
This is debatable in my opinion. Regenerator allows you to switch around more freely and helps mitigate the damage from entry hazards but you need to actually switch out to activate its effects, meaning you can't just sit there stalling a pokemon or perhaps you risk losing momentum trying to heal your pokemon. Recover doesn't help you switch in easier but it does heal more health and can be used whenever you want, meaning you don't have to switch out making it preferable for stat-up sweepers. They are both very good but they work differently so I don't think they're comparable.
Well, that depends. Drizzle is better in most situations, but if you want to abuse Swift Swimmers then RD is preferable for obvious reasons. Not to mention that RD can be stuck on any pokemon in order to change the weather--whether it be to start up rain again if Politoed goes down or to forcibly change the weather against certain teams (although you'd probably run Hail for that--and you don't have to risk bring one pokemon in over and over to nullify opposing weather. So yes, I'd say Drizzle is better but Rain Dance has its merits.
Again, this kind of depends. On a stat up sweeper, it'd go Shed Skin > Refresh > Natural Cure in my opinion. A wall would want the abilities, but Heal Bell or Aromatherapy would be preferable to Refresh any day of the weak and, depending on your team, those two abilities as well.
Pure Power/ Huge Power
Mummy can actually come back to bite you in the ass if you're not careful and since it only activates on certain attacks you can't invoke it whenever you wish. As for Mold Breaker vs Gastro Acid, Mold Breaker again is preferable in singles but Gastro Acid has the additional benefit of helping your team get rid of annoying abilities (did you see that triples warstory TheMaskedNitpicker posted the other day? It showcases how Gastro Acid could be effective there as you could see by his removing Truant from his Slaking and also crippling an enemy Machamp). Also consider that Mold Breaker only nullifies abilities that prevent certain attacks, whereas Gastro Acid erases ALL abilities. Mold Breaker will not prevent that opposing Haxorus from EQing your Bronzong; Gastro Acid will. Mold Breaker will not stop that Gliscor from stalling you with Poison Heal; Gastro Acid will. Mold Breaker will not stop that CM Reuniclus from shrugging off Toxic; Gastro Acid will. Et cetra et cetra.
Moody I suppose, but again I would contend that Acupressure does have an advantage in doubles in that you can use it on your allies, not to mention it doesn't carry the risk of lowering your stats (can't end up with +2 acc -1 Def, or +2 Sp. Att -1 Acc, etc etc)
Trace, unless you're playing multibattles and you want to guarantee you get the right ability.
That kinda depends. Unaware will negate the advantage of heavily-boosted pokemon, but if they kill your Unaware pokemon all their boosts are back and you're SOL, whereas Haze forces them to start over no matter what, even if you eat it. It also destroys BP chains better than Unaware due to Ingrain.
Intimidate
Bad Dreams
I would say Work Up. Download gives you an instant boost, yes, but what you get is out of your control and it's either one or the other. Work Up allows you to increase both stats at once which I would say is better for mixed sweepers, not to mention you can use it more than once for increased power whereas with Download you get one boost once and that's it.
Pressure though they both suck.
Sturdy
The abilities, I would assume, although you are adding in another variable that is allowing those abilities to shine (if the weather gets nullified all the moves instantly become better than the abilities) so it is possible that auto-weather itself could be a confound.
So yes, in most cases, the abilities are better than the moves, although I would say many of the moves have their merits. We also run into the problem that we have a potential confound with auto-weather, and for Sand Veil/ Snow Cloak specifically we have to determine if it activates so often and is so detrimental to the metagame to potentially screw over a bunch of pokemon with said ability by banning them.