With a family , I would stay first 4-5 days in Paris, get over jet lag, rent an apartment, see sites and do a daytrips( Versailles maybe?),, then hop in rental van,, drive to somewhere else, i]spend a week or two there, and so on and so on. I am not a fan of moving along every two or three days,, especially with younger kids.., they will make and keep memories longer if they actually get to stay a bit in a place. I know some people like to rush about and collect places they have "done" but I think it takes at least a week in any country to get a proper taste.
As for where to go,, that is not something anyone can really answer for you and your family.. What do you all enjoy,, canoeing in the Dordogne sounds fun to me,, seeing Mont St Michel,, swimming in the Med. , touring the ruins in Rome?
I personally think kids should see Paris and Rome, ( hey even a 5 year old can find the history of the Coliseum fasinating) and most kids learn all about the French revolution,, I think history is great for kids, so a mix of history and physical fun trips sounds like a good mix to me.