(Especially if you're rich, so you could afford a boarding school—the closest real life equivalent I can think of to fostering the child with another lord.)
I'm a guy so.....
This world is a very different one to Westeros. Westeros would be more easily compared with Medieval England. In this world, marrying a complete stranger who you barely know is frowned upon by most people and is considered antiquated. Cat herself says she didn't mind him cheating as they barely knew each other and a man has needs. Also in this day and age we live in a world where every man is considered equal, very unlike Westerosi society, and so could expect a measure of their fathers inheritance and what not. Bastards in Westeros wouldn't
Yes bringing home Jon is a bit of a slap in the face for Catelyn but I do think that Jon had a right to be there if his father so chose. I personally think Ned should have put his foot down and told Catelyn where she should stick it when she told him he wasn't welcome at Winterfell without Ned there.
We don't know what Daemon was like or how his relationship with his legitimate bretheren were. Theon Greyjoy wanted to be a son and impress his father. Blood is thicker than water. Jon wanted Winterfell but is honourable and feels guilty for having wanted Winterfell. He obviously loves his brother and I very much doubt he would do anything to try to take his claim or harm him. He was a fast learner and had/has the potential to be an able battle commander and so would have been a lot more useful to Robb in the field than his Lady Mother who I feel is personally responsible for his downfall.