If I have Eternatus VMAX with the "Eternal Zone" Ability in play, can my opponent use Mawile GX's "Captivating Wink" Ability to play any non-Darkness Pokémon onto my bench?
No, if Eternal Zone is working, your opponent cannot force you to put any non-Darkness Pokémon into play.
Source: TPCi Rules Team (2020-09-24)
When using Mawile GX's "Captivating Wink" Ability to put Pokemon onto my opponent's Bench, are they allowed to evolve those Pokemon on their next turn since they just came into play?
Yes, as long as it's not the opponent's first turn of the game they can evolve those Pokemon if they are able to do so. It is a basic game rule that neither player can normally evolve any Pokemon on their first turn of the game, even if they are going second.
Source: TPCi Rules Team (2020-03-12)
What happens if I use Steelix's "Gigaton Shake" attack, and then my opponent uses Mawile GX's "Captivating Wink" Ability during their turn? Would those Pokemon I put into play be able to attack on my next turn (since it was not during MY turn)?
No, they would not be able to attack.
Source: Unified Minds FAQ (2019-07-19), TPCi Rules Team (2019-08-01)
Can you use Mawile GX's "Captivating Wink" Ability if your opponent's Bench is full?
No. It's public knowledge that you can't put down any additional Basic Pokemon on your opponent's Bench, so you cannot use the Ability.
Source: Unified Minds FAQ (2019-07-19), TPCi Rules Team (2019-08-01)
Mawile GX's "Captivating Wink" Ability says to put any number of Basic Pokemon you find in your opponent's hand onto their Bench. Does this allow you to put more than 5 Pokemon on the bench?
No, Captivating Wink does not allow you to exceed the normal limits of the game. You can put any number of Basic Pokemon onto your opponent's Bench up to the maximum size of the Bench.
Source: Unified Minds FAQ (2019-07-19), TPCi Rules Team (2019-08-01)