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)
You can play Bellelba & Brycen-Man as long as there's at least 1 card in either player's deck. It can also be played by discarding 3 cards from your hand if at least 1 player has more than 3 Benched Pokemon. But if at least 1 player doesn't have more than 3 Benched Pokemon, you cannot discard cards from your hand.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)
You can do any part of Red & Blue as long as there's a Pokemon in play that can evolve into a Pokemon-GX AND there's at least 1 card in your deck.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)
If I try to play a Stadium card but it is discarded by the effect of Chaotic Swell stadium, can I play another one?
No, it was immediately discarded without effect, but you already played a Stadium card which can only be done once per turn.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)
When I use Roxie to discard two Pokemon that each have the "Blow-Away Bomb" Ability, do those abilities stack so I can place two damage counters on each of my opponent's Pokemon?
Yes, each of the Pokemon gets to use their own "Blow-Away Bomb" Ability when discarded by Roxie.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)
You can play Cynthia & Caitlin as long as there's a Supporter card in your discard pile. You can also play it by discarding a card from your hand if there's at least 1 card in your deck. But if you have no cards in your deck, you cannot discard a card from your hand.
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)