New puppy woes!

We have a new 14-15 week old puppy. He was sold to us by someone who bought him from a breeder. They said they were selling him because he was going to be too small for their 3-year-old who was being too rough with him. That may be true. BUT---

To put it bluntly, he is a peeing and pooping machine. He will "piddle" outside 3-4 times while I watch him walk around on the lawn. Each time he goes I give him a tiny treat. He walks back in and that little stinker can squeeze out another drop or two within a few minutes. He can also lay out a tootsie roll even if he has done that outside. If I can sit down with him after he comes in I can catch him and take him outside again, but sometimes that is hard. If I don't have a leash on him all the time he will run a few steps away and--well, you know!

I have put him in a medium sized kennel with his bed and a piddle pad. He goes on the pad. I have confined him in a small kennel that has been made so he would have to sit in his urine if he piddled--he piddled anyway! My friend told me that a dog trainer suggested that with her difficult-to-train dog.

Is he just trying to mark his territory? I have two other dogs who are housebroken, but I can't say that they have never had an accident.

He is not mean, but he wants to play a lot more than the older dogs and they are having a rough adjustment. I feel like Sadie is saying, "Mom, how could you do this to us?" My children told me not to get another dog even though he's the cutest little beast you've ever seen--but I didn't listen to them anymore than they used to listen to me. I guess I have become a teenager again and they have become the mom!

HELP!