Question

What are all the abilities that prevents status conditions and stat reducing moves?

Answer 1

All Stat Reducing Moves are prevented by White Smoke and Clear Body
*Hyper Cutter protects from decreased Attack stat only
Defiant boots a Pokemon's Attack when any stat is decreased, and this has the side effect of not allowing their Attack to be decreased.
Speed Boost Pokemon can still have its Speed decreased, but it won't be long before it is back to normal (or faster than normal), so it is protected from Speed drops, in a fashion.
$tarfire also reminded me of two more:
Big Pecks protects from its Defence being lowered.
Keen Eye prevents its accuracy being lowered.

As for preventing statuses:
Water Veil protects from Burn
Limber protects from Paralysis
Vital Spirit and Insomnia protect from Sleep
Magma Armor protects from frozen
Immunity protects from Poison
Oblivious protects from Attract
Own Tempo protect from Confusion

Poison Heal will heal the Pokemon when it is poisoned

Magic Bounce will reflect any status move back at the attacker, but doesn't protect from statuses, per se as it can still be affected by moves with a chance to inflict a status

Magic Guard allows the Pokemon to be inflicted with a status, but it won't lose health from being Poisoned or Burned.

Natural Cure - it can still be inflicted with a status, but it will be healed when it switches out

Here are another couple $tarfire added:
Hydration cures statuses while it is raining.
Leaf Guard cures statuses while it is sunny.
Inner Focus protects from flinching.

I think that is everything - if I missed something let me know in the comments, and I will add it into my answer.

Comments

Big Pecks and Keen Eye

Hydration and Leaf Guard (Don't forget to credit me :3)

And last but not least Inner Focus.

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thanks

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i thought about those and wasn't sure about if they counted, but I'll edit them in. And good point :)

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Speed Boost and Defiant shouldn't be on here because it doesn't technically prevent stat drops.

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technically, they don't. however, the net effect is the same as if they prevented stat drops so i added them. I will take them out if you really want, but you could argue Magic Guard, Magic Bounce, etc. shouldn't be on here either.

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Leaf guard and hydration are not the same effect. lol

Answer 2

Update for Gen 8

New Abilities relating to Status Conditions
Guts- Attack is boosted when Burned, Paralyzed, or Poisoned. Negates Burn effects.
Quick Feet- Speed is boosted when Burned, Paralyzed, or Poisoned. Negates speed drop from Paralysis.
Tangled Feet- When the user is confused, it's Evasion is raised by 50%.
Marvel Scale- (not sure if this counts) Boosts Defence when Burned, Paralyzed, or Poisoned.
Flare Boost- Special Attack is boosted when Burned.
Toxic Boost- Attack is boosted when Poisoned.
Water Bubble- Prevents Burns.
Shed Skin- User has a 33% chance to remove any status conditions at the end of each turn. It's not presented from being inflicted, but they can be healed.
Comatose- Cannot be inflicted with any major status conditions.
Healer- Again, doesn't prevent the user from being inflicted, but there is a 30% chance to heal an ally's status condition at the end of every turn that the user is on the field.
Early Bird, Insomnia, Vital Spirit- All of them prevent Sleep.
Sweet Veil- Prevents the user and ally Pokémon from falling asleep.
Aroma Veil- Prevents anything from limiting move choices, such as Taunt, Torment, Cursed Body, etc. (Not sure if this counts)
Pastel Veil- Prevents user and allies from being Poisoned.

Abilities relating to Stats
Flower Veil- All Grass-type Pokémon on the field cannot have their stats lowered or be inflicted with status conditions.
Full Metal Body- Prevents any stat reduction.