In the book, Nick Dunne talks a lot about his father and mentions several times that he wants to be a better father to his kids. In the movie, the investigation of Amy is ended by the detective, but in the book it is Nick who ends it after he finds out about the pregnancy. The female investigator is actually really disappointed that he no longer want to pursue the investigation. So you are right, staying with Amy has everything to do with his father.

He is sure it is his baby because she used the his sperm sample that she saved to get pregnant. The movie makes it seem like this takes place just a few days after she returns, but in the book it actually occurs several months later. Since so much time had passed, he would know that it wasn't from someone else, unless of course, she snuck out without him knowing.