Quaquaval ex

If Iron Hands ex with Heavy Baton is KO'd by Quaquaval ex's "Exciting Dance" attack which pushes it to the Bench after doing damage, does Heavy Baton move energies attached to Iron Hands ex since it is on the bench by the time KO's are checked?

Yes. Heavy Baton checks for two things here: Iron Hands ex was damaged by an attack while it was your Active Pokemon, and Iron Hands ex was Knocked Out by that attack. So even though you don't check for a KO until the attack is completed, the criteria for using Heavy Baton has been met, so it can be used. This holds true for any similar Ability that triggers when the Pokemon is Active and gets damaged by an attack.

If Radiant Jirachi with the "Entrusted Wishes" Ability is KO'd by Quaquaval ex's "Exciting Dance" attack which pushes it to the Bench after doing damage, do you still get to search your deck for up to 3 cards or not since Radiant Jirachi is on the bench by the time KO's are checked?

Yes. Entrusted Wishes checks for two things here: Radiant Jirachi was damaged by an attack while it was your Active Pokemon, and Radiant Jirachi was Knocked Out by that attack. So even though you don't check for a KO until the attack is completed, the criteria for using Entrusted Wishes has been met, so it can be used. This holds true for any similar Ability that triggers when the Pokemon is Active and gets damaged by an attack.

If my opponent flipped heads for Nymble's "Slight Splash" attack on their previous turn, what happens if I use Quaquaval ex's "Exciting Dance" attack? Does Nymble get switched out to the bench or not?

The effect of Exciting Dance is done to the protected Nymble, so Exciting Dance would not move it to the bench.

My opponent's Active Pokemon is Eternatus VMAX with the "Eternal Zone" Ability working, and they have 8 Pokemon on their Bench. I use Quaquaval ex's "Exciting Dance" attack, doing damage to Eternatus VMAX and then sending out Ting-Lu ex. Will Ting-Lu ex's "Cursed Land" Ability shut off "Eternal Zone" before the entire effect of "Exciting Dance" completes?

Yes. The switch effect of Quaquaval ex happens, and then with Ting-Lu ex entering the Active Spot, Eternatus VMAX's Ability gets shut off, meaning that player must immediately discard Pokemon until they have 5 Benched Pokemon. Then, Eternatus VMAX switches with a Benched Pokemon.