France Itinerary suggestions

I would consider staying on the eastern side the entire trip. Meaning, replace Normandy and Loire with an additional stop or stops between the alps, burgundy and Alsace. There’s plenty to see in those regions. This also is a favor to your next trip where you can dedicate your time to some combination of Normandy, Brittany, Loire and Dordogne without crossing over to the east.

Is this your first trip to France? You've got a couple of long driving days. St Remy to Amboise with a couple of pit stops, not sightseeing stops, will be a really long day.

You have a good reasonable plan to see the varied areas of France. You do have long transit days from Nice to St Remy and St Remy to Amboise, so those days are mostly free of sightseeing. Here is your trip as I see it, just listed slightly differently. You have planned well to get a taste of 4 areas in France so bravo for you! It might be wise to plan on even more time on the road. One never knows!
Day 1 arrival-sleep Nice
Day 2&3 visit Nice & area -sleep Nice
Day4 drive to St Remy de Provence (3-5 hour drive)-sleep St Remy
Day5, 6, 7 visit St Remy and area-sleep St Remy
Day 8 drive to Amboise (7-9 hours plus another hour for gas/food/comfort stops)-sleep Amboise
Day 9, 10 visit area (chateaux)-sleep Amboise
Day 11 drive to Bayeux (4 hour drive)-sleep Bayeux
Day 12th & 13 visit Bayeux area-sleep Bayeux
Day 14 drive to Caen (30 minutes), drop car and take train to Paris-sleep Paris
Day 15 & 16 Paris-sleep Paris
Day 17 fly home

I like to check out drive times on Viamichelin.com. They often give a choice of routes (e.g. fastest, most economical, most scenic, etc.). Just remember to add in time to gas up, eat, etc. Likewise with Google maps. Neither one seems to account for roadworks, etc.
When you decide the things most important for you to see and experience, check back here if you would like tips, hints, suggestions from forum members. You appear to have made good lodging choices as well. (We often stay with relatives, so I don't have a lot of experience re hotels, especially around Paris and in Provence). Sorry!
You will have a wonderful time! Bonne chance!

We stayed at the Grand Hotel du Luxembourg in Bayeux and liked it very much. And Bayeux is such a nice town, we really enjoyed walking around. The Cathedral was definitely worth a visit. Also the Bayeaux Tapestry, but I think I may have heard that it is being cleaned or restored so won't be available.

We also enjoyed Amboise. We visited the Chateau du Close Luce (Leonardo da Vinci's house), and Chenonceau, which was just beautiful.

Sounds like a lot of driving, but if you don't mind that, it is a pretty good plan. (I HATE driving or even riding in the car, so I'd be on the train for all these trips!)

I haven't been to Bayeux yet, but all the others are among my favorite places in France (and Bayeux is on my bucket list!)

Thank you for your replies and suggestions. If we decide to change our plans to Provence, Alps, Burgundy and Paris and skip the other region, which specific cities/areas would you recommend for lodging. Is Nice worth the visit?

Just fyi, Nice is on the Riviera, not really in Provence, but adjacent. Factor in the distances.