Really good (and difficult) questions... I'm starting to think about this from the MtF perspective. Currently I'm "dad," and my initial feeling was that, despite discomfort with a lot of things, I had grown to identify with that name/family role. After thinking about it for a while, though, I do think I'll eventually change to something else tbd. A big reason is the discomfort I imagine would come from being called by an obviously wrong-gendered name when out and about.
My wife and I are still thinking about this, but we plan to roll out the changes pretty slowly (kids are 1-8 years old now) so it'll be a while before we have to deal with this. I think we'll let the kids decide what they're comfortable with, but encourage them to shift away from "dad." I think the younger kids won't have a problem, but it may take the older some time. But I'm confident that the kids will adapt, and I bet yours will too!