So when I bring Ash Greninja over to Moon from the Demo, can I use a Synchronize Abra to get it Modest Nature? Or is it randomized? It will then be traded over to USUM but I don't think that affects it.
Yes it is possible to use synchronise Abra on Ash greninja
Whether greninja is adamant, timid or modest, Greninja is still a gift Pokémon and can have his nature changed by Abra.