France Itinerary

You've got a good outline for a fast-paced but feasible trip, I have suggestions to make it more relaxed further down.
But first, sticking with your original plan, I would reverse the order for logistical and weather reasons: head straight to Nice from overseas (flying open jaw) and end up in Paris. That way you'll save a preflight night in Paris at the end, and you'll have hopefully better May weather up north, whereas April is already good down south.
Route would be:
- Nice area 3 nights, no car needed
- Drive to Arles, 4 nights
- Optional overnight in Carcassonne
- Drive to Dordogne area e.g. Sarlat, 3 nights
- Drive to Loire area e.g. Amboise, 3 nights
- Optional overnight in Mont Saint Michel
- Drive to Bayeux, 2-3 nights (to me this is enough, 1 day beaches, 1 day tapestry and Caen memorial)
- drive to Caen then Train to Paris (or drop car off in Bayeux if possible), 4-5 nights

But as you can probably see, it's rushed. Personally I would either
- drop the Riviera / Nice area, and head straight to Arles instead. (fly to Marseille from overseas)
- or drop Dordogne, after Arles drop the car and take the train to Tours, then rent a new car for Loire and Normandy
You save three nights that can be put to good use to slow down the pace of the trip.

I'd recommend more than 2 or 3 nights in the Dodogne. It takes time to get there, and there are many beautiful villages (and caves with prehistoric art, etc.). To me it's at least a 5-night sort of place.

When your itinerary includes areas that are really far apart, you need to figure it will take you just about a full day to move to the next place, so 3 nights is only 2 days.

France is a huge country, and I wouldn't recommend trying to cover such a large part of it in just 3 weeks.

I'm not a fan of chilly, wet weather, and I am doubtful anout the Dordogne in April (I was there in June) and Normandy in May (I was there in July).

I agree with the previous comments, and would recommend omitting one of the three: Dordogne, Loire Valley, or Normandy. Each of those three is spread out and will take a lot of time to see. I think you will regret it if you try to cram them all in, because you'll be spending most of your time and energy checking in and out of lodgings and navigating hours-long road journeys from one region to the next.

We did a similar road trip last Sept, 22 nights but just 2 in Paris as we’ve been there before.
I’d give more time to Dordogne/Sarlat we did 5 nights, it was our favorite area- so much to see and do there
Loire was our least favorite area

We left car in Arles and took train to Nice, flew home from Nice
You might look at open jaw flights to avoid getting back to Paris or reverse trip as suggested above.

We actually really liked Nice, liked Arles a lot, also stayed StRemy which is a great base if you have a car, loved Sarlat, loved Bayeux

I'd leave out the Riviera but include the Rhone Valley are of Provence (e.g., Arles and environs). You could easily spend a week in either Provence or the Dordogne, or both if you are willing to give up one of your other destinations.

The fast-paced version is Paris > Normandy > Loire > Provence > Dordogne, flying home from either Toulouse or Bordeaux. (You can also reverse it, ending in Paris). EG:

Paris 3 nights, train to Bayeux, rent car no. 1.
Bayeux 3 nights (or last night at Mt St Michelle), drive to Amboise
Amboise 2 nights, TGV to Arles, rent car no 2
Arles, St Remy, or other 6 nights, drive to Carcassonne
Carcassonne 1 night, drive to Sarlat or Beynac or other
Sarlat 6 nights; drive to airport in Bordeaux or Toulouse for afternoon flight home.

The above would benefit from more time or a willingness to dispense with either Normandy or Provence. There are lots of terrific destination in SW France.

About the Dordogne: it's a real gem, hard to get to, and you probably won't be back soon. Resist the temptation to cut it short; better to save for another trip.

Thanks to all for your suggestions. Will modify our trip based on your input. We definitely do not want to feel like we are spending all of our time traveling.

I would be thinking more of an open jaw; in to Paris and out of Marseilles. From Arles its only a 52 minute drive to MRS Airport. Drop the car off at the airport. That would save you the long and $$$ train back to Paris.