Yveltal with "Fright Night" Ability knocks out Wobbuffet with the "Bide Barricade" Ability which was turning "Fright Night" off. Does Exp. Share on the Benched Pokemon behind Wobbuffet work?
Yes, since Wobbuffet was not yet in the Discard Pile, Fright Night remains off and Exp. Share works.
Say I have an Active Pokemon with Energy Pouch attached and a Benched Pokemon with Exp. Share attached. If the Active Pokemon is KO'd, how do the effects of the tools resolve?
You may choose the order in which the tools' effects resolve. There is no preference of one tool over the other in this situation.
Can I play Target Whistle and choose a Basic Pokemon in the Discard Pile with the Omega Barrier Ancient Trait, or does Omega Barrier block the Target Whistle?
Yes, you can put that Pokemon from the Discard Pile onto the Bench. Ancient Traits are only active when that Pokemon is in play.
If I play Captivating Poke Puff and find a Pokemon with the Omega Barrier Ancient Trait in my opponent's hand, can I choose that Pokemon to be benched or does Omega Barrier block the Captivating Poke Puff?
Yes, you can put that Pokemon onto the Bench. Ancient Traits are only active when that Pokemon is in play.
If I use Evosoda to evolve my Basic Pokemon into a Stage 1, can I then use Wally to further evolve it into a Stage 2 on that same turn.
Yes, but be careful of the order in which you play the cards. Wally says you can use it on a Pokemon that was put into play this turn, but Evosoda says you cannot use it on a Pokemon that was put into play this turn. So you can play Evosoda first then Wally, but you cannot play Wally then Evosoda (on the same Pokemon, of course).
If my Yveltal with "Fright Night" Ability gets KO'd, am I then able to use Exp.Share on one of my Benched Pokemon to recover one of Yveltal's Basic Energy cards?
Sorry, but no. Fright Night would still be working so Exp.Share would not enable before Yveltal hits the Discard Pile.