If I use Xurkitree GX's "Lighting GX" attack to put a card from my opponent's hand face-down as one of their prize cards, should we mix up the prizes so he doesn't know which one it is or does it stay wherever I put it?

Rearranging the prize cards is not part of the attack, so wherever you put the new prize card is where it should stay until some other action causes the prize cards to be rearranged.

Relevant cards

SMP_SM68