If my opponent used Quaxly's "Apply Gel" attack against Tinkatuff on their turn, could I use Tinkatuff's "Pulverizing Press" attack and ignore the requirement to flip?
No, as long as the Apply Gel effect is on Tinkatuff you would have to flip heads in order to even start your attack.
Source:
Paldea Evolved FAQ; TPCi Rules Team (2023-06-08)
If Annihilape ex only has 10HP left, can I use its "Angry Grudge" attack and place all 12 damage counters onto it?
Sure, ka-boom!!! You can put up to 12 damage counters on it even if it doesn't have that much HP remaining. Annihilape ex will do 20x each damage counter that was placed, and at the end of the attack Annihilape ex will be knocked out.
Source:
Paldea Evolved FAQ; TPCi Rules Team (2023-06-08)
Can you use Manaphy's "Pulling Currents" attack when your opponent has a full bench?
It's public knowledge that you can't put down any additional Basic Pokemon on your opponent's full Bench, so although the attack can be used you cannot look at your opponent's hand and you cannot put any additional Basic Pokemon on your opponent's Bench.
If my opponent has Duraludon VMAX with the "Skyscraper" Ability in play, when I use Inteleon VMAX's "G-Max Spiral" attack and choose to return the only Special Energy card attached to Inteleon VMAX to my hand, does "Skyscraper" prevent the damage or not?
The choice to remove the Special Energy card is made first, but then damage is done and finally effects of the attack. So removing the Special Energy card is an effect that doesn't happen until damage has been done and therefore blocked by Skyscraper.
If my Lumineon V attacks with "Aqua Return" to take my last prize card, but Lumineon V is my only Pokemon in play, do I win because I took my last prize before shuffling Lumineon V into my deck or do I lose because Lumineon V get shuffled in before taking my last prize?
Everything has to resolve before you check for win conditions, and in this scenario both players have achieved one win condition; the game would then be resolved by either Sudden Death or a Tiebreaker Game. However, if the opponent also has no remaining Benched Pokemon then they would lose due to 2 win conditions vs. only 1!
If I have 6 cards in my hand and use Mimikyu's "Impersonation" attack to discard Erika's Hospitality from my hand, can I use the effect of Erika's Hospitality?
Since you have 5 cards in your hand after discarding the Erika's Hospitality you can use its effect.