Provence/Côte d'Azur with teen girls

If you can, add Grasse, where perfume was created-what could be more fantastic for a teen girl.

I would definitely try to get to the Camargue to see the flamingoes and wild horses.
I really liked the Chagall museum in Nice.
It's always beach season.

Sarah,
To get to Avignon you don't take a train from Gare Du Nord. Those trains leave from Gare de Lyon.
Trains and buses from Nice will get you to many places along the Riviera coast. I would return my car after dropping my bags at my lodging there. It's been years for me, so I can't answer if Nice or Villefranche-sur-Mer is better. I do know parking is a big pain in this area, and not cheap. You can even get to Monaco by public transportation, and many do. To be honest, I wouldn't take the time to go to Monaco with only two full days on the Riviera (3 nights=2 full days).

Re perfume...Grasse is a great town to explore the creation of perfume and its history. It is on the way from Moustiers-Sainte-Marie to Nice, which is about 3 hours drive, so you could stop there for a tour of a perfumerie and have lunch, still getting to the Nice area not too late in the afternoon.
Are the teenage daughters late movers in the mornings (like most teenagers)? If not, you should be able to do all you have planned for each day, with the exception of combining Arles with the Camargue. That sounds too rushed to me. I could be wrong about that. You should allow time to find your way into the various cities and towns (some are harder to navigate than others), locating parking, and just getting to where you are heading, be it lodging or a sightseeing venue. Build in some time for traffic tie-ups, etc. You will have less stress and more fun. If it turns out that you have time to spare, all the better. Have a coffee or aperitif somewhere and relish being in France!

Sarah...Another thought. As you decide on activities, etc. do verify days and hours of operation for them, just in case they have limited hours and days. Kayaking in the Gorge Du Verdon may have some closures if the current is too strong in the spring. I haven't been river rafting in some years (I am now 82), but I recall some outfits here I n California that opened later than usual in the season because the runoff was too large and too strong, and the river was unsafe. Pay attention, also, to any place that may require reservations ahead of time, although except for the kayaking maybe, I don't see that necessary in your case.

Some comments on your questions:
* In Avignon, most see the Palace and the bridge. From Avignon, people tend to go to the Pont du Gard and Nimes.
* Going to the Carmague may be a lot in one day. There are a lot of sites in Arles. Nearby St. Remy is also Les Baux, and you might find the Carrieres du Lumieres to be interesting.

* Besides Gordes, Rousillon is worthwhile with a nice walk through the petit Colorado.

* Outside Aix is Lourmarin, a very nice village. Avoid rush hour on the ring roads.
* Moustiers is pretty small but you can hit some neighboring villages as well. The drive on the north side is more spectacular, and there are trails there as well.

* In Villefranche-sur-Mer, you car decision may depend on your accommodation. In the lower village you have access to the beach and the train station, so it is pretty easy to get around, but parking can be a problem. In the upper levels, where some apartments are located, you will need a car or walk a lot of steps. Be watchful of when the cruise ships dock -- they can quickly overwhelm the village in the days between when the first tenders arrive and the swarms depart. (Many immediate head to Monaco or Nice or tours, others hang out).

* If you drive to Eze, there are branches of the perfume factory for Fragonard there. You can continue up to La Turbie and the Trophee des Alpes (or all the way up la Tete du Chien, for even higher views... there is a small trail at the end). But the perfume experience in Grasse is better. Past Grasse are villages like Tourettes sur Loup and Gourdon, plus on the return route either Mougins (via Cannes) or via Vence/St. Paul de Vence.
* I have not taken the palace tour but the wealth is most prominent in Monte Carlo, near the casino and Hotel de Paris (and adjacent shopping). Near Villefranche sur Mer is the Villa Ephrussi and the expansive wealth of Cap Ferrat, hidden behind gates and fences. If you enjoy walking, you can walk around Cap Ferrat (or Cap de Nice). You may wish to head into Nice, as there may be more for the teens there.
* Antibes has some nice beaches on the Gulf side, but allow for traffic getting into the town. It is easier to reach by train.