Larvitar

My Larvitar is poisoned with 10 HP left. My opponent has one prize remaining and no cards in the deck. Can I attack with "Mountain Eater" and win the game?

You can attack, but if you can't discard the card you don't get to remove two damage counters. Larvitar will be KO'd between turns and your opponent will take the last prize.

Say my opponent and I each have 1 prize card left, then I use an attack like Pumpkaboo's "Reckless Charge" that KO's my opponent's Active Pokemon but also KO's Pumpkaboo due to "Reckless Charge's" recoil effect. However, my opponent has only one benched Pokemon with 20HP left, while I only have Iron Valiant ex on my bench. Does our game go to a tiebreaker since we both have a Pokemon on the bench, or will I win the game since Iron Valiant ex would be promoted as Active and then "Tachyon Bits" will be activated resulting in the KO of my opponent's last Pokemon?

All effects have to finish resolving before the winner is determined, so you could activate Tachyon Bits to KO the opponent's last Pokemon. You would win the game as you have achieved two win conditions but your opponent has only achieved one.