I just need clarification/confirmation, really. The overlay page says "The image stays in your bucket forever, and you can assign it an infinite number of times!" so does this mean if I "unassign" a given pet's overlay, and adopt that pet out, I can then reassign it to a different pet? In other words I'll still have the overlay available and can put it on whoever I want?

Yes, that's what it means :)

Okay, good. Thank you!