One month in France in july (help needed !)

I would pick three or four places and rent an apartment or AirBNB or something for a week in each one. Choose according to your interests. My choices would be Paris, Brittany or Normandy, and Nimes/Avignon, but that doesn't give you any mountains. Schedule them in a circle if you are flying in and out of Paris to cut down on the travel time between locations.

I always think of weather first. Normandy and Brittany give you excellent odds of avoiding miserable heat. A car is very handy there.

I think there are attractive outdoor/nature activities for you in the Dordogne (caves, canoeing, walking), but I don't know what the weather would be like in the middle of the summer. The area was wet and not particularly hot for me in late May/early June.

Normandy/Brittany and the Dordogne are very far apart, though.

Provence is often oppressively hot in the summer. Annecy can be bad as well. If you choose that area, be sure you have air conditioned lodgings.

Return to Alsace ??

I think if it were me, I would choose western France and perhaps do a mixture - you could start in southern Brittany and include Vannes and Carnac, then go to Nantes (with the very cool Machines de l'Île, a big hit with kids), Puy du Fou (which I am jealous; I have wanted to go there for years!), and then you'd have some other options - Loire valley / castles (and maybe stay someplace fun like a troglodyte house), or possibly Île de Ré for another kind of atmosphere. It won't be mountainous at all, but if you also like the seaside, you'd have several options, along with fun things for kids to do and see.

If your final itinerary takes you to Brittany, consider Village Gaulois in Lannion.
It's a throw-back kids' amusement park where lots of the games are played with ropes and pulleys. Parents hand-crank the merry-go-round for toddlers, and pull the chariots and barrel trains that children ride in. Boats are paddled (with canoe paddles) around a lagoon firing nerf balls at each another (or streams of water when we were there). A hand-powered catapult, a zipline close to the ground, that children ran back to do again and again. Historical crafts demonstrated, such as basket-weaving and sword-making.
We chanced upon it while looking for something else and were delighted. We (two grandparents with no kids along) spent several hours there and had a good time. Imagine kids (and parents) having fun with no electronics. That's what we saw when we were there.
I'm not doing it justice, so take a look at this review from TripAdvisor:
https://www.tripadvisor.com/Attraction_Review-g1067714-d1977913-Reviews-Le_Village_Gaulois-Pleumeur_Bodou_Cotes_d_Armor_Brittany.html

Wherever you stay, be sure it has AC. July and August can be extremely hot.

Crime isn't a valid reason to skip Marseille - the crime rate is fairly standard where tourists are likely to go - but summer heat is, and there's nothing to entertain kids there (the beaches aren't great).

Chamonix is great for a bit of the Alps (more like Banff, Jasper than the Dolomites). I think the Côte d'Azur most anywhere will be very crowded and expensive and not at all good with a car. Maybe Lake Geneva - it has beaches and it's in the area of Chamonix/Annecy.

Aside from those areas, I've only been to châteaux and Alsace . . . I started from Chamonix and stopped in Cluny for a night on the way to Bourgogne, finished in Strasbourg. Lots of charming villages, farmers' markets, a few châteaux and abbeys, and the kids would love Monkey Mountain in Alsace (interact with a colony of Barbary macaques). I spent 4 nights in Avalon and explored western Bourgogne, stopped in Beaune for 1/2 day and then 3 nights in Eguisheim (just outside Colmar), ended with 2 nights in Strasbourg, where I dropped the car.