As we've shared on another thread: if you stay in Nice, take a taxi to the airport; if you stay in another resort, spend your last night in a Nice airport hotel. We are based in Cannes when we travel, and it's very stressful to take a taxi and know you will arrive in time for a 6 AM flight.. From someplace other than Nice, you have to factor in your travel time to the airport, and from Cannes, the farthest distance on your list, we allow an hour. We discovered the Novotel Arenas and stopped doing that after our first few trips.
Whether to keep your car while your are in the Riviera is really a matter of choice. A car isn't necessary as the public transportation system in this part of France is convenient and efficient, and parking and traffic can be an annoyance. I also think that public transporttion is part of the culture that I enjoy, but then, I live in Kentucky where things are spread out and we don't have similar transport...we drive and I like not doing that.
Nice will be lovely, but if you want beach time and would prefer something softer than Nice's "stoney" surface, I'll suggest again, look towards Cannes, Antibes and Juan les Pins. However, the natural surface in Nice maybe a novelty in your trip vs. the sand that is imported to the others. In the end, however, you won't go wrong with whatever base you.choose. The are all lovely, and each one is an easy train or bus ride away from the others.
You are at a part of your planning where everything is on the table, and it will be less overwhelming once you've narrowed things down. You will put together the trip that works best for your family.