Say my opponent uses Trevenant & Dusknoir GX's "Pale Moon GX" attack on my Keldeo GX with "Pure Heart" while Power Plant stadium is in play. If I replace the Power Plant with another stadium during my next turn, does Keldeo GX get knocked out at the end of the turn or not?
If Keldeo GX remains the Active Pokemon it will be Knocked Out. Keldeo GX's Pure Heart Ability *prevents* effects, but it does not *remove* existing effects. If an attack like Pale Moon GX places an effect directly on Keldeo GX, that effect cannot be removed by reactivating Pure Heart; in other words, effects that are removed when returning to the Bench can't be removed by Pure Heart.
Source: TPCi Rules Team (2020-01-09), TPCi Rules Team (2020-02-06)
Say I have Keldeo GX already in play, and my opponent uses Latios GX's "Clear Vision GX" attack with Power Plant in play. If I play a different Stadium to get rid of Power Plant, can I use Keldeo-GX's "Resolute Blade GX" attack now?
Yes, you can. When Power Plant is removed Keldeo GX's Pure Heart Ability turns back on, and Keldeo GX is no longer hit by the global effect of Clear Vision GX. If an attack does not place an effect on Keldeo GX directly, like an attack that puts an effect that isn't removed when returning to the Bench, it can be removed by reactivating Pure Heart.
Source: TPCi Rules Team (2020-02-06)
Can you use the "Dedechange" Ability when you play Dedenne GX from your hand while Power Plant is in play?
No, you cannot. Dedechange does not activate until Dedenne GX is on the Bench, at which time it is considered "in play" and therefore blocked by Power Plant.
Source: TPCi Rules Team (2019-08-22)
Can you use Greninja GX's "Elusive Master" Ability while Power Plant is in play?
Yes, you can. You get to put Greninja GX onto your bench AND draw three cards because Elusive Master activates while it is in your hand before it comes into play, and therefore is not blocked by Power Plant.
Source: TPCi Rules Team (2019-08-22)
Can you play Darkrai GX with "Restoration" Ability from your Discard Pile if Power Plant is in play? And if so, do you do all of its effects?
Since the Discard Pile is not considered "in play", Darkrai GX can use its Restoration Ability to put it onto the Bench and attach a Darkness Energy card from the Discard Pile to it. Once all of the Ability's effects are complete, then Power Plant shuts off the Ability.
Source: TPCi Rules Team (2019-06-06)
If I have Pokemon GX with Abilities in play and the Power Plant Stadium is also play, can I now play Green's Exploration since those Pokemon "have no Abilities"?
Yes, you would be able to play Green's Exploration in that case.
Source: Unbroken Bonds FAQ (2019-04-19), TPCi Rules Team (2019-05-02)