Riviera in July. Car or Train? Also, what are the most "French" parts?

It will be very hot and humid. Whether or not to rent a car depends on your itinerary. Are you going city to city or exploring the countryside. We rented a car for our two weeks in Provence and the Cote d’ Azur. The train would have limited us. That being said we did take buses from time to time to get to certain places such as Monaco and Cap Ferat when we were based in Villefranche sur Mer for example where a car would have been a hindrance in our travels (parking and traffic).

The train provides good service along the coast from the Italian border to a point just east of St. Tropez. For most of the hill
towns, you would take a bus or need a rental car. So it depends a lot on what places you want to see, Parking is likely to be difficult and costly in most of the most attractive places in that area, so you shouldn't rent a car for days when it is not needed.

Rick's guide book to southern France has good coverage of the area and will allow you to narrow down your list of destinations.

Yes, it will be hot. At that time of year, the nearest not-hot places are the Basque country or high in the Alps or Pyrenees.

I have my doubts about the legality of sleeping in a vehicle outside a campground, but I have never researched that.

As for most French, I think that will be difficult since northern Europeans (from the UK to Russia) pour onto the Riviera in the summer. With that goal, you would do best with a car up in the hills, visiting places whose names you haven't encountered up to this point.

What do you mean by "French"? The Riviera has been a major tourist spot for a good century, both for foreigners and people from other regions of France. If you want stereotypically "unspoilt", you should probably pick a different region.

July is a pretty warm and crowded month. If you are staying along the coast, you can easily navigate using public transport (except for the St. Tropez area). A car is really best for the hills and to reach the smaller, lesser known villages. However, outside of St. Paul de Vence, the better art museums in the area tend to be in the coastal areas, in places like Nice.

As you might expect, you'll see a heavy Italian influence the farther east you go. If by "French", you mean without this influence, you need to head further West (starting from Nice). If you simply mean less international, you need to go to the inland areas. Even then, there is a Mediterranean influence throughout the South. This Mediterranean influence is part of the Regional culture, and is reflected in the food, architecture, and even the mannerisms of the people (even in Western Provence). Now, if you are simply looking for a higher percentage of French tourists, as you get to the western edge, you may find more French holiday goers, especially in the 2nd half of July.