I am so glad to see they are conceding ownership. Rather he ever shows up or not, I would think that your talk with her in Target would be reason enough to just assume that he is now your puppy dog. How fitting that you were in there shopping for dog stuff & she was looking for baby stuff! Just shows that you have more time & resources to take care of him at this time than they do. I'm not even saying they are wrong, of course, our human babies take precedence but it is the right thing for them to let him go to someone that has more time for him right now.
I wouldn't feel obligated to pay them for any of his things but if you offered that would be kind of making it more of a formal transfer of ownership.
Surely, they wouldn't accept any money since you are giving them baby things but even if they did, it would be worth it to say, Yea he's mine. If I did give them anything, I'd make it a check and on the memo line write "Rusty & things" just in case.
I'm sure none of that is necessary & you may never hear any more about it. She'll probably just go home & tell her husband, boyfriend, whatever?? that she saw the neighbor & told her that they were willing for Rusty to stay with you since he didn't seem to want to come home anyway.
