What happens if a Pokemon with Splash Energy attached is Knocked Out by an attack while Lost City stadium is in play?
The order of resolution for the effects can be decided by the owner of the Pokemon with Splash Energy attached to it. So the Pokemon can either go to the Lost Zone and stay there, or use Splash Energy to return the Pokemon to your hand.
Can I play Single Strike Mustard if my Bench is full?
No, you cannot play the card. Since the Bench is full you cannot search your deck, and since you did not search your deck you do not get to draw 5 cards.
If my Snorlax is affected by its "Thumping Snore" attack but then on my opponent's turn they use an effect that puts Snorlax to sleep, does the 2 coin flip effect of "Thumping Snore" get replaced by the regular 1 coin flip effect during Pokemon Checkup?
The newer sleep condition would replace Thumping Snore’s sleep condition, so you would now only have to flip 1 coin instead of 2.
If I use Malamar's "Synchro Kinesis" attack and the only card that each of us have in common are Basic Energy cards, do I get to do the extra damage or not?
If you and your opponent both have Basic Energy of the same type in your hands, it's a match. Professor's Research and Boss's Orders with different characters on them also count as matches.
Source:
Lost Origin FAQ; TPCi Rules Team (2022-09-08)
Do I have to have Tool cards in my discard pile in order to use Rotom V's "Scrap Short" attack? Or if I have Tool cards in my discard pile, do I have to choose at least one of them to put into the Lost Zone?
You can use Scrap Short if you don't have any Tool cards in your discard pile. Scrap Short does not require you to put any Tool cards into the Lost Zone; if you cannot or choose not to put any Tool cards into the Lost Zone it only does the base amount of damage.
Source:
Lost Origin FAQ; TPCi Rules Team (2022-09-08)