Paris/Provence/Nice -Gorges du Verdon ?

With 10 travel days, I would skip Provence and base your family in Paris for half the time and Nice the other half. There is more than enough things to do to fill the time. We are visiting France this July with our 13-year old and they are looking forward to seeing the catacombs, taking a “hidden Paris” bike tour, going up to the top of the Eiffel Tower, thrift store shopping in the Maris, and a macaron baking (and eating!) class. In Nice we doing a walking food tour that ends with a full meal, and taking day trips to Cannes (to see the international fireworks show on July 14), Antibes, Menton and maybe Eze Village. So much to do!

Your plan is doable. We love Paris and could spend weeks there without thinking twice. We recently did this trip you are talking about doing in August. We spent two nights in Avignon at the Bristol Hotel. You can take the train from Paris to Avignon for your two nights. We stayed at the Bristol Hotel. It was reasonable, and a short walk from the train station. Actually a lovely street with restaurants and shops, and walkable to just about everything. We liked the train station locale for day trips. L' Isle Sur La Sorgue is a pretty town you can train to. They have a big market on Sundays. You can also train to Arles and Nimes from Avignon. You would need a car for St. Remy and the Luberon, or you can take the bus. If you have a car you can drive to Nice, but Nice itself is not car friendly. You could turn the car in at Avignon, and train to Nice. We stayed at https://all.accor.com/hotel/9907/index.en.shtml the Ibis Styles Centre Gare hotel which is a very short walk from the Nice Ville train station. We picked this hotel because of its excellent reviews, and the fact that it was a short walk to the train station for day trips. While some areas around the train station aren't the prettiest, this was a decent location. Nicely decorated rooms, clean, and decent size. It also had some of the most comfortable hotel beds we have ever slept on. Also had a great breakfast, and the staff is awesome. Easy walk to beach and old town Nice. Your teens will enjoy day-tripping to Monaco, Cannes, Antibes, Menton, Eze Village, Ventimiglia, Italy, and so much more. The suggestion of above of skipping Provence isn't a bad one either. The Riveria will be crowded for sure (and hot!) Always be sure and book hotels that have AC. Not every hotel does.

I just returned from Avignon and stayed at the Hotel de l'Horloge. It was very comfortable, and reasonably priced with great service. I booked a room with a terrace which I very much enjoyed. The location is perfect for visiting the Papal Palace and other major sites. I ate at the restaurant "EAT" which was very good and within walking distance of the hotel. Just make sure to make reservations for dinner. Train between Paris and Avignon was fast and an easy trip.

In Paris I stayed at the Hotel Molière in the 1st which I really liked. Great breakfast and service. When I stay in the 7th I am a fan of Hotel Muguet. If you are going this summer, just be prepared for large crowds in Paris. I told my husband I had PTSD from the museums in Paris, I have never seen them so crowded before!! Just go with an attitude to be flexible since I found the crowds overwhelming. Two places I ate at that I really liked were Le Recamier (specializing in soufflés with a nice garden setting) and Le Café de Mars near Rue Cler (they have live music on Saturdays at lunch which was very festive).