If I have strange Gizmo (After forging a key destroy each creature and each artifact) in play and a soul snatcher (Each time a creature is destroyed, its owner gains 1[A])

After forging a key I get to decide timings as I am the active player, I assume I can do the timings so that I gain the Amber from my own creatures being destroyed, but can I make it so that the soul snatcher goes out of play after my creatures, but before the opponents creatures, thereby denying them the amber?

you're not getting any aember, all creatures and artifacts are destroyed simultaneously by Strange Gizmo. (And neither is your opponent by the way)