Must the child be purified, along with the bathroom and whatever the child touched, if he touched his private parts after relieving himself?

As long as you know that you are afflicted with severe waswasah (whispers of Satan/obsessive doubt), you must disregard it and push it away, for "the basic principle concerning our things is purity." So, whatever you are certain is impure, then that is it. But whatever you doubt, the basic principle is its purity. Do not tire yourself and your family by indulging in doubts.