Spiritomb

After using Unown VSTAR's "Star Cipher" attack while Path to the Peak is in play, I then use Spiritomb's "Ticking Terror" attack to change the Defending Pokemon's Weakness to . If my opponent removes Path to the Peak from play, what will that Pokemon's Weakness be?

The Weakness will be . Once Path to the Peak is removed from play Star Cipher's effect will reapply, and the most recent Weakness-changing effect takes precedent.

My Arceus V is active and I have Dunsparce with "Mysterious Nest" Ability on my bench. If my opponent uses Spritomb's "Ticking Terror" attack does Arceus V now have weakness, or does "Mysterious Nest" keep it with no weakness?

The Ticking Terror attack would switch Arceus V's weakness to , but then Dunsparce's Mysterious Nest would change Arceus V back to no weakness. However, since Ticking Terror says "Until the end of your next turn..." Arceus V would become weak to if Dunsparce were to leave play or have its Mysterious Nest ability shut off.

If I use Spiritomb's "Ticking Terror" attack against the Defending Pokemon and their owner moves it to the Bench during their next turn, does its Weakness return to normal?

Yes, since it is an attack effect, moving that Pokemon to the Bench removes it.

If my opponent has no Weakness and I use Spiritomb's "Ticking Terror" attack, will I do extra damage on my next turn due to Weakness?

If a Pokemon has no Weakness printed on the card, Ticking Terror will give that Pokemon a x2 Weakness to Darkness.

If I use Gengar's "Cursed Drop" attack, can I put damage counters on more than one of my opponent's Pokemon, or do I have to put all four counters on one Pokemon?

You can put the 4 damage counters on their Pokemon in any way that you like.