Yes. According to https://www.viamichelin.com/ it is 02h41, 225 km. That assumes no stops or delays, so add 25%.
The end of the road is just past Lauterbrunnen, at Stechelberg. This is the bottom of the cable car up Schilthorn, and you can park there.
The quickest route is via Bern and past Thunersee (Lake Thun). The alternative route is via Luzern, the Brünig pass and Brienzersee (Lake Brienz). Viamichelin estimates this at 03h07, 239 km. I would say this route is more scenic, you could go one way and return the other.
Yes, you could drive there in the late afternoon/early evening. But than you would get to Lauterbrunnen late. I assume you are staying in Lauterbrunnen, and going up to Mürren, Gimmelwald and Schilthorn on following days.
You will need to buy a "Vignette", cost CHF 40, at the border. This is a road toll for using Autobahns in Switzerland. You may be lucky and the car already has been to Switzerland in 2018 and already has a 2018 Vignette.
I assume you are returning to France afterwards. Renting a car in France and returning it in Switzerland has a high surcharge.