Paris? I usually start and finish there so at least 2 or 3 nights on the front end and another similar stay at the back end. Since there is always plenty to see and do in and around Paris, you can always add any extra days you end up with.
Britanny or Normandy? I'd much rather go to Britanny and I think your kids will enjoy it more. In the summer, you can visit a Pardon (traditional festival). My favorite places in the north were Cancale, St. Malo (the kids would love the hike to the islands at low tide) and Dinan (great old center and cafes on the port). I really like Ft. La Latte. It's an old fortress that has served many purposes over the years but now shows as the medieval castle it once was. There are places to see out west but they are mostly wild coast. Locronan is a really pretty town but I'm not sure there is much there for the kids? My favorites in the south were Auray and Vannes (maybe add Port Aven). Seeing the prehistoric ruins around Carnac is reachable from either. Vannes has an incredible old center with ramparts and a large port area (sailboats not industrial). Auray is a much smaller version of Vannes. My favorite prehistoric sight is Cairn du Gavrinis, a passage tomb similar to Newgrange in Ireland. It's only accessible by a small ferry from Larmor Baden. The tour is all in French, which won't be an issue for you.
Dordogne or Provence? In summer I'd prefer to stay out of the heat of Provence. I'd suggest considering Lot Valley or the Languedoc area as a less crowded alternative. Otherwise, Dordogne is beautiful. I prefer the hills, oak forests, rivers and green to Provence's arid/hot desert feeling. Save Provence for cooler weather in the fall. I went mid-September and it still felt too warm.
Alps or Pyrenees? At first Brittany was my favorite place in France, then Burgundy, now it's the Languedoc area (Occitane). That area has the same green hills and medieval towns as Burgundy but they are less populated and more spread out. People in the area apologized and said the Spanish side was sunnier than the French side, but the French side is green while the Spanish side resembles the coastal mountains in California.
In this case, you could put Brittany, Dordogne, Lot, and Pyrenees together seamlessly into a vacation.