Car in Nice??

Rent the car in Nimes and take the train back to Nice.
I experienced the traffic jams just taking the bus in Nice around the area of interest for tourists: the Promenade Anglais and the pedestrian only old town.
You only need a car to get to Les Baux and the Camargue.

Drop the car at Nice Airport and continue by taxi or tram, as there is a stop right at the airport that goes into Nice and the old town. Trains to Monaco and Menton go every 15 minutes. You can either take the No 1 tram to the main station, or a local bus to Nice Riquier station.

Oh Canada,
Just a hint...with five adults be sure to rent a car big enough for all of you AND your luggage. The descriptions of car sizes on the rental sites are not deliberately deceptive, but the cars are not as roomy as we are used to here in North America. You may want a small van, especially if you drive to Nice from Cassis or Nimes.

Stop at Saint Paul de Vence on your way back to Nice. If the tourist train is running to Tende, you can drop the car at the airport after Saint Paul and take the train up to the Mercantour for your day trip.

If the train isn't running, keep the car for one night and go to the Parc du Mercantour (Tende) the next day and then return the car to the airport. If you are going to the Vallée des Merveilles, you need to reserve a guide and the 4x4s. You can't drive in there yourself.

I've driven in Nice in summer and lived to tell the tale. Patience and the smallest car that suits your needs.

Just be wary of driving all the way to Nimes from the NCE airport on your first day. You can pick up the train directly at the airport to say Avignon or Nimes, and pick up a car from there. Fortunately it appears you will miss the Monaco Grand Prix.

I too recommend leaving the car behind, and hitting St. Paul before you do. The train line to Tende is under renovation right now through September, I believe. A bus replaces the route through the spiral tunnels. See the cautions on booking ahead, if you intend to hike into the Vallee des Merveilles. If you do keep a car an extra day, you might consider staying in the Carre d'Or, Carabacel/Wilson/Medecin area, as these hotels and apartments may have better access to parking and AC, compared to the Old Town, but are just across the Promenade du Paillon. Note in the Port, the UN Oceans Conference starts on the 9th of June.

If you do stay in Nice, there is an express exit that runs under the city out to the main autoroute, via tunnel Andre Liataud Ientrance is just off the Promenade des Anglais and the corner of the Old Town, on the inner lanes heading north). It is not obvious on Google maps. The last day you hold the car you can simply return it back to the airport. From the airport returns center at P5, you can head to terminal 2 and pickup the tram.

Thanks everyone for great suggestions and advice. We did rent a small van. Leaning towards staying 2 nights in St-Paul and 3 in Nice (without car) after reading your comments.