Yes, Normandy and Brittany tend to be cool and overcast even in mid-summer. There might be the occasional hot spell, but unbearable heat isn't likely to be a long-lasting phenomenon there. For a July trip, I think time in those two regions is a brilliant idea. Yes, there may well be occasional rain, but you will very much appreciate the temperatures. Since I think you'll be more rain-tolerant after the (likely) heat to the south has wiped you out for a while, I'd try not to make Normandy/Brittany my first stop.
The TGVs (fast trains) in France tend to fan out from Paris. You can get all the way to Avignon in about 3 hours, Aix in less than 3-1/2 hours, and Bordeaux in less than 4 hours. Nice doesn't have TGV service, so that trip from Paris takes considerably longer. For the faster-to-reach destinations, flying isn't advantageous unless you can book a multi-city trip that takes you right to a southern destination as part of your transatlantic ticket (perhaps returning home from Paris). That worked for me in 2017, when I wanted to start my trip in Nice.
Something to keep in mind about the French rail infrastructure: It's easier to travel roughly north to south (or vice versa) near the eastern or western edges of the country than to cross the country from east to west or v.v. You can burn a lot of time with the latter, so try not to do that repeatedly. Also, you may find yourself needing to change trains in Paris on trips that don't--from a map--seem to require you to visit that city. That's the lines fanning out from Paris bit. If that proves necessary, pay attention to the details, because you may well have to change stations in Paris. It's not the end of the world, but it's annoying, especially if you are burdened with much luggage and a 7-year-old. Be sure you choose a departure with a transfer time that allows you to get to the new station by Metro, RER, bus, taxi or foot.
I have no children, so I'm not the one to suggest child-friendly activities. I'd just say to read hotel/lodging reviews carefully to be sure people don't complain about the quality of the air conditioning.
