When I use Arceus & Dialga & Palkia GX's "Altered Creation GX" attack do I take an extra Prize card only if I Knock Out that Active Pokemon with an attack, or do I take an extra Prize for ANY of my opponent's Active Pokemon that get KO'd by my attacks during the rest of the game?

You get to take an extra prize if your Pokemon's attack Knocks Out ANY of your opponent's Active Pokemon during the rest of the game, not just the one that was the Defending Pokemon at the time of the Altered Creation GX attack.

Relevant cards

SM12_156

SM12_220

SM12_221

SM12_258