I think they can put it on whenever if it's in their closet from before you took it down. I adopted Julian from someone and he sometimes wears a Totoro hoodie which defiantly didn't come from my able's. He had it on the first day he was out of boxes which was a little surprising but I think it looks really cute on him so I don't mind
I haven't had that issue but I've had a villager from another island come with a custom design. It's probably stuck in their cycle right now until you gift them more clothes to take it out