If Klefki with the "Wonder Lock" Ability is attached to a Pokemon as a Tool card, what happens if "Silent Lab" stadium comes into play? Does "Wonder Lock" shut off and Klefki get discarded, or does it remain attached to the Pokemon?
Once Klefki is attached as a Pokemon Tool card it retains its effect even when its Ability has been disabled. Once it's attached, it is essentially treated as a Tool with that text, so shutting off its Ability doesn't matter at that point. However, if something such as Lysandre Labs is in effect, Klefki won't be discarded at the end of your opponent's turn.
If Lysandre Labs is in play and I use Primal Groudon's "Gaia Volcano" attack to discard the stadium and do enough damage for KO, what happens if the full HP Fighting-type Defending Pokemon has Focus Sash attached to it?
While Lysandre Labs is in play Focus Sash has no effect. However, Lysandre Labs is discarded as part of the attack before KO's are checked. So the Focus Sash would be in effect and if all of its conditions are met the Defending Pokemon is not KO'd but has 10HP remaining.
If Lysandre Labs is in play, do effects that depend on the presence of Tool cards, such as the damage increase of Doublade's "Curse of the Blade" attack or the energy cost modification of Rotom's "Roto Motor" Ability still work?
If an effect depends simply on the presence of a Tool card in a certain location, the effect still works. It is the Tool card's effects that are cancelled, not effects based on the presence of a Tool.
Source:
Forbidden Light FAQ (2018-04-20), TPCi Rules Team (2018-05-03)