You have 9 places listed for 23 days, yes, quite ambitious. Some of your first nights are after a lot of driving, so those days are spent in a car. Whatever Michelin or Google gives for driving time, you need to add a lot more for eating, rest stops, getting lost, etc. Every time you change hotels, you loose a half day immediately.
You have Nice, Menton, and Monaco along the Riviera. No need for a car which is difficult to park in this area. Take the train. No need to change hotels between Nice and Menton since they are only 30 minutes from each other. Monaco is three train stops before Menton, about 10 minutes. I'm not sure why you have 2 nights for Nice but 3 for Menton. There's a lot more available to see and do from Nice than Menton. I have family living next to Menton so know both Nice and Menton quite well. Do you want to see anything else in this area?
Chamonix and Lauterbrunnen are outliers that will take a lot of time and effort to reach. You will have only two days in each place because your first night will be after those tiring drives. With only two days, any bad weather and there go your views of and from the mountains. Have you checked to see if lifts will be running in the off-season?
I would compact your itinerary to Bordeaux, Nice, Strasbourg, and Paris, dividing your time in these four with day trips.
I haven't researched how to get from Nice to Strasbourg, whether by train or plane. I'm not even sure you'll need to rent a car, except for what the previous poster suggested, since you should be able to find organized trips to St. Emilion from Bordeaux and the Riviera has the train service that runs every 20 minutes.
I wouldn't worry about rain in Nice in October; it's a Mediterranean climate. It could have a downpour now and again but there's also change of season fog and humidity in the air that adds to precipitation. I live in the same climate a few hours from there. Nice is by no means just summer destination; it's France's 5th largest city with a population of 348,000. That doesn't include the megapolis along the coast consisting of all the other towns where people live year round. Nice is a major city that gets tourists all year round.
It's so beautiful landing by plane at the Nice airport. Lucky you if you fly from Bordeaux.