Welcome to the board, pattyparker64!
If you have the time to spend with him, and he's bonded to you, it's not cruel to keep him. But do you think he's a dog from your neighborhood that's just been allowed to roam? )You say he "appeared on your doorstep time after time"....) Someone may be missing him. At the very least, you should check with the local shelter to see if anyone has reported him missing, and put up posters or ask around your neighborhood to see if anyone has been actively looking for him.
If he
does end up staying, you'll need to start thinking about exercise for the little guy. If there is a dog park near you, that might be an answer--but you'll have to figure out how socialized he is, first (some dogs don't get along with other dogs). Leash-walking is an option, too--and good for both of you!
Do some browsing for indoor games you can play with your dogs--if you stimulate his mind, he's less likely to get into mischief. There's no reason that you and he can't make each other very happy!