Is it legal for a 17-year-old to date a 20-year-old and move in with him and his parents and not get anyone into trouble?

No, the minimum age of majority is 18, except for Alabama and Nebraska where it is 19, Mississippi and Pennsylvania where it is 21. If the parents of the young man allow the minor to live in their home w/o the permission of the girl's parents they as well as their son, can be subject to criminal and civil penalties.