Being a woman who is very much aware of such cases like this (though, no affair innocent child involved nor a person in public office) through personal experience, I'm thankful that both Elizabeth Edwards and John Edwards came to a place of understanding between then and, most likely, for their children.
I won't go into personal details and being older now can understand/respect what this family has evolved to. Playing devil's advocate again: Perhaps healing and forgiveness for past behaviours has brought this family together. Perhaps marriage proposal has everything to do with being a family again in a new form. None of us know the dynamics of the behind the private scenes of their lives. Nor should we. As long as all are working it out as best they can at this time, who are we to judge?
I still think the man was an idiot, but I don't believe he would publically, or privately, hurt his children with some not thought out decision and have them part of it.
As far as what his children decide to call his wife to be, that's completely up to them. I have a step mother who is 5 years older than me. I do not call her "Mom". Would be kinda creepy if I did. Blech....
FWIW, the woman who married my father 2 years after the passing of my Mother is not the person he was involved with. I want to be fair to her about that.