Lyon is a large city with a lot of varied sights. You are proposing to go there for just one full day and a few hours on the day you arrive from Paris--unless you plan a late departure to Annecy. Lyon is worth a lot more time than you are giving it. Interesting as it is, it would be a good place for a longer stay.
Annecy is beautiful, and with luck your timing will protect from the swarms of tourists I encountered there in June 2017. But I struggle to imagine spending six full days there.
Along the Riviera a great deal can be accomplished by train. The regional-train line runs along the coast, stopping at essentially every town down at beach level except for St. Tropez. A car is not helpful in seeing the coastal towns--quite the contrary. Parking will probably be frustrating, time-consuming and expensive. Sone of the small hill towns can be reached fairly easily (if not super quickly) by bus; those include Eze-Village, St-Paul-de-Vence and Vence. Buses tend to fan out from Nice, though there are exceptions.
There are different opinions about the best place to stay along the Riviera. As an art fan, I favor Nice, which has at least six art museums. Logistically, it's hard to beat. Others prefer the feel of a smaller town, and Villefranche-sur-Mer and Antibes both have adherents.
Unless your trip is so late that you encounter low-season rates, I suspect you'll find room rates in Nice more reasonable than in the favorite smaller towns. Nice has a lot of hotels, so it's a competitive market. However, I have only researched rates in Nice, not in places like Menton, Villefranche-sur-Mer and Antibes.
There are certainly a lot of places you can visit in Provence without a car--Aix-en-Provence, Arles, Avignon, L'Isle-sur-la-Sorge, St-Remy-de-Provence, Nimes and Orange among them. But it's not so easy (and sometimes impossible) to get to the tiny villages you may have read about without a car, unless you want to sign up for one-day bus tours. Once a decision is made to rent a car (note that rates have been very high this year, so best to check), the selection of a base comes down to considering the driving time to all the places you hope to see. Aix-en-Provence is an outlier to the east, so wouldn't usually be recommended. Quite a few people on the forum like to stay in St-Remy, which is small enough that it doesn't have a train station, just bus service from places like Arles and Avignon. You can use ViaMichelin.com to get estimated driving times.
The Riviera and Provence are different experiences. I'd say the Riviera has more art while Provence has more Roman ruins. There's more open space in Provence. For me (a non-renter of cars in Europe) a major difference lies in how much easier it is to visit the near-coastal part of the Riviera without a car. But that's not going to be an issue for you.
For weather data I mostly go to the website timeanddate.com. It shows actual, historical, day-by-day weather data for most places of any size. Here are Annecy's weather statistics for September 2012. To get a sense of how much rain might be expected, I use the climate chart included in most cities' Wikipedia entries. Those charts usually also show monthly average hours of sunlight.
Since you mentioned being in Europe for "almost three months", I want to be sure you are aware of the 90-day limit in the group of countries constituting the Schengen zone. If you're spending some time in the UK, Ireland or the southern and eastern Balkans, you will be fine. If your entire trip will have you in the western part of continental Europe, you need to be very careful not to go over the 90-day limit. Both your arrival day and departure day count. Exceeding 90 days may well get you a substantial fine and result in being banned for years.
