If you use Peonia and put a Treasure Energy or Dream Ball from your prizes into your hand, can you use the Treasure Energy or Dream Ball effect to attach it to one of your Pokemon?
Sorry, but Treasure Energy and Dream Ball only trigger when you are taking it as a face-down prize card, not when you are trading it with another card.
Source: TPCi Rules Team (2021-10-07); TPCi Rules Team (2022-04-07)
When using Peonia, do the cards I take count as "Prizes taken" for effects that count how many Prize cards you have taken?
No, it does not.
Source: Chilling Reign FAQ; TPCi Rules Team (2021-06-17)
When using Peonia, do I have to randomize my Prize cards after putting them back?
Rearranging the prize cards is not part of Peonia's effect, so wherever you put the new prize card is where it should stay until some other action causes the prize cards to be rearranged.
Source: Chilling Reign FAQ; TPCi Rules Team (2021-06-17)
Can I play Peonia if there are no other cards in my hand?
Sure. In that case, you still put up to three Prize cards into your hand and then put them back down.
Source: Chilling Reign FAQ; TPCi Rules Team (2021-06-17)
If I choose to take a face up Prize card with Peonia, do I have to put its replacement prize face down or face up?
The replacement prize goes back face down.
Source: Chilling Reign FAQ; TPCi Rules Team (2021-06-17)
Do I get to look at the Prize cards I'm drawing with Peonia before choosing them?
No you may not, unless the Prize cards are already face up.
Source: Chilling Reign FAQ; TPCi Rules Team (2021-06-17)