Kian gave you excellent advice, and I must praise you for being willing to actually hear it, and come up with reasonable questions in response to it. Clearly you care, and I have to agree that returning him to a place where no parents were to be seen, and this pup was already ripped from his Mum at 5 weeks, is no place to which he should be returned.
He is as cute as a button, but as Kian said, that won't help in the long run. May I also suggest you sit your daughter down and teach her right now about when and how to purchase a puppy!!! If she is old enough to write a cheque, she is old enough to learn this now. She has made this whole situation your issue with which to deal, which is quite unfair. My heart goes out to you.
Obviously if you opt to keep this puppy, he will be your life for at least the next six months to a year! You will have to enroll in obedience classes and perhaps even hire a trainer after that. All of these things cost money and time. I'm glad you are able to try and start potty training, although realistically there is no hope of that happening until at least 20 weeks. You will be trained, not the puppy as he can't even begin to give you a signal as HE doesn't even know when he has to go.
Even if you are not going to keep the puppy, I would still strongly suggest a visit to the Vet. The wee fellow deserves at least that. Perhaps your Vet can help you re-home the dog, if that is your choice, or you can use "Petfinder" to help you. Personally, I would want to know precisely where this pup was going, and to know that the people who take him are prepared to give him his loving, safe, FOREVER home, for his sake.
Best wishes to you both!