You said 10 days but you’ve done ten nights which is eleven days. When and how are you getting back to Paris? What time is your flight from Paris.
In my view you are trying to do tooooo much.
Doable yes but what do you do in Nice or Chamonix for 3 days ? To truely appreciate the Alps you need to drive. If you just go to Chamonix, it's like a single tourist attraction. Do the train, do the cable car, that's it your day is gone. You go to the Alps either to hike or to go on a road trip. Your itinerary just feels like ticking boxes for a first time French trip.
You go to Nice for the villages, the mountains, the coast, it all takes time. And Chamonix alone is certainly not worth the trip from India. For example if you want to see a glacier, the Glacier de l'Argentière is better than Mer de Glace, which is retreating too fast, it's just an S of gravel right now with some white appeaing here and there. Other parts of the Alps offer better glacier viewing opportunities : the hike to Glacier Blanc near Briançon, the cable car to the Meije glaciers from La Grave etc... All of this to say that with only 10 days, I would choose one or the other. It's a long trip to either Nice or Chamonix from Paris... There is no direct fast train, Nice is 6h from Paris, and Chamonix, I guess you have to go to Geneva and take a bus, so it's even longer. That's 2 days gone out of your 10 day itinerary.
For Paris + the French Alps, the best way is the TGV to Grenoble (3h from Paris). In the 5-6 days you'd have, I'd rent a car, check out La Grave and the cable car to the glaciers, then drive to Briançon, visit the city, check out the wild Vallée de la Clarée, with the mélèzes starting to get orange in autumn, if you're hikers, do the hike to Glacier Blanc from Pré de Madame Carle (go to the refuge du glacier blanc, and climb a bit higher to get a better view, but don't get on the glacier), then if you have time visit a bit of Queyras, like Château Queyras, Fort Dauphin, the historic village of Saint Véran, maybe a nice easy hike to Lac de Sainte Anne. Go back to La Grave, hike the Lac Noir and Lac Lérié, which are gorgeous, and go back to Grenoble and Paris.
O yes. 11 days it is.
I haven't booked my flights yet. I wanted to finalize itinerary a bit. The reason I've mentioned Paris is due to cheaper return tickets.
Well like I said my answer is that to truly experience France, you should do Paris and then a small road trip on the side. With so little time there is certainly no time for 2 destinations besides Paris. The TGV lets you reach a few destinations from where you can do a 5-6 day road trip : Avignon, Grenoble, Strasbourg, Lyon, Rennes, Bordeaux. Respectively they let you explore regions like Provence, the Alps, Alsace, Beaujolais, Brittany/Normandy or Dordogne. Maybe you're scared driving in France if you only know about driving in India, but my guess is if you can drive in India and stay alive, you can drive anywhere. Switching driving side on the road is just a small adjustment, you just follow other people and the signs and you'll be fine.