I agree with the others--the scope is too large for a 2-week trip. Two issues not specifically addressed so far:
Two days in Avignon (and it's really just one full day plus some hours) is totally inadequate for seeing Provence. It's enough time to Avignon and maybe one other, easy-to-reach spot. Way too high a ratio of travel time to feet-on-the-ground time. I think Provence is worth at least five nights, and if you don't have a car you'll still miss out on a lot. With less time, what are you going to skip? Arles? St-Remy-de-Provence? L'Isle-sur-la-Sorgues? Nimes? villages like Roussillon?
I haven't been to Eze so I can't comment on its charms. But I don't think it's a good base for a bunch of day-trips (and Eze itself is too small to keep you occupied for long) unless you are going to rent a car and the places you want to see are up in the hills. You could certainly plan a trip like that (just be sure your lodging in Eze can provide reliable parking guidance), but do you not want to see some of the coastal towns? Are you going to skip places like Nice, Villefranche-sur-Mer, Menton, Antibes, Cannes. Cap Ferrat and Monte Carlo? (I am not saying they are all musts; I myself skipped Cannes and Monte Carlo.) Those places are well served by the regional train running along the coast, and driving to them wouldn't be smart because of parking challenges and costs. Eze has only bus service, so for day-trips to the coastal towns you'd have to take a taxi or bus down to the coast and connect to a train. That would not be an efficient way to see the coastal towns.