If I attach Island Challenge Amulet to a Pokemon-GX that has less than 100 HP left and it's Knocked Out, does my opponent take only 1 prize?
No. You only get the reduction in Prize Cards taken if it is Knocked Out by damage from an opponent's attack.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)
Can I play Acro Bike if there's only one card left in my deck? And if so, do I put that one card into my hand or discard it, or do I get to choose?
You can play Acro Bike with only one card left in your deck; but you have to put it into your hand, you cannot discard it.
Source: TPCi Rules Team (2019-10-31)
If my Lillie's Poke Doll was Knocked Out by my opponent, can I use Oricorio GX's "Dance of Tribute" Ability?
Yes, even though they didn't take a prize, Lillie's Poke Doll was still considered a Knock Out of one of your Pokemon.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)
You can play Mallow & Lana as long as you have a Benched Pokemon. But if your Active Pokemon is not damaged, you cannot discard 2 cards from your hand.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)
Can I use the second part of Misty & Lorelei to discard 5 cards if I have not already used my GX attack for the game?
No, you cannot. If you haven't used your GX attack yet, you can't discard 5 cards from your hand.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)
When I play Misty & Lorelei, can I discard one or more of the 3 {W} energy I searched for as part of the 5 cards discarded to get the re-use of my GX attack on this turn?
No. You have to discard 5 cards "when you play this card". That is before you search for the {W} energy cards.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)
Can you choose the order of effects when playing a Tag Team Supporter, or do you have to do them in the order stated on the card?
You must do them in the order stated on the card. However, you don't have to meet the criteria for the first effect in order to do the second effect; for example, if the first effect says to search your deck but you don't have any cards in your deck you can still do the second effect as long as you meet its criteria.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)
You can play Misty & Lorelei for the base effect as long as there's at least 1 card in your deck. You can always play it by discarding 5 cards from your hand.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)