I haven't watched the episodes since they aired, and I wasn't paying the closest attention to that plotline at the time, so I can't speak to TV Robb's thought processes. But I think that in terms of the books, it's worth considering that he elected not to risk a bastard child precisely because he had seen the awkward effects of Jon's upbringing at Winterfell on both Jon and Catelyn, and didn't want to repeat those unfortunate circumstances.
That didn't seem to be the thought process of television Robb to me, that he was acting out of any concern for how Jon was raised and the effects on Ned and Catelyn's marriage. He seems like a man who thinks he's very much in love but is in reality making a kneejerk decision based on his heightened emotional state--which is, I suppose, what the book Robb who's a teenager trying to save the honor of a high born girl he fucked and is in fear of her being pregnant is doing as well. That business of the Westerling girl and a "honey trap" set while she's nursing his wounds sounds like angsty good fun. I may have to read the books.
I'm not sure whether I want Robb's wife to be devious and complicit or on the straight and narrow and genuinely in love with him. The former might be better for creating story.
So he's supposed to be the same age as Jon, huh? Not buying that at all. Kit Harrington comes across very much as a young, sad, confused bastard kid limited by his life choices but Richard Madden, though I think he's a fine actor, comes across as a poised, older and much more mature individual, especially when he's dealing with Ned's men and his soldiers. Jon seems like a kid while Robb comes across as a man.
I never thought about the situation with King Robert, but he would demand the death of any remaining Targaereyan child. Excellent point. It makes sense that Ned would keep that secret to save his sister's child, even from his wife to protect the secret to his grave. Robert spoke of sending someone to kill Danearys back in episode 2, while Ned looked thoroughly uncomfortable and rejected the idea.