Never plan to return a rental car just before a flight; never be more than a cab ride away from the airport the night before a flight. Car rentals are not efficient operations like in the US -- it can take an hour to get a car and it can take awhile to get a receipt when you turn it in and you want that receipt to prove it was returned gassed up and undamaged. And all sorts of things can get in the way -- traffic, car breakdown, slow rental car processing of meeting time deadlines.
For a flight to the US you need to be in line checking in 3 hours ahead; we have had situations where that was barely enough and others where we had time for coffee after security; it is not entirely predictable. I'd book an airport hotel for the last night and plan to come in fairly late but I would not be rushing for the airport that morning. You need to be in line at 10 for a 1 pm flight and that is the busy time of the day at airports.
You didn't ask but spending 2 days in each of 3 very distant spots is going to make this a trip mostly about getting there with no a whole lot of being there. With so little time, you would be better off with 3 at the most locations closer together e.g. Burgundy and the Loire or even Riviera and Provence and then Paris. You would also be well advised to fly open jaw if you want to do this and finish your trip in Paris rather than beginning there. Even if you have round trip tickets in and out of Paris you would be better off taking the train to Nice your first day and beginning the trip there and ending in Paris so you don't waste your last day rushing back to home base. Your arrival day is sort of miserable jet lag day so use that one to get to your most distant point and you don't waste the day before last rushing back for the flight. Learned this the hard way.