Ankur,
For the trip from Paris to Chamonix, you might consider something like this......
- Travel via high speed TGV from Paris / Gare de Lyon to Geneva-Cornavin (about 3 hours).
- Travel via pre-arranged Shuttle from Geneva to Chamonix (about an hour).
This will likely be a faster option than travelling the entire route by rail.
Based on the official Chamonix schedules, it appears that the lifts and other facilities will be working on 15 May. Be sure to budget accordingly as the trip to the top is not cheap. One other point to mention is that the top level of the Aguille facility is at almost 13,000 feet / 3842 M. so if you have any issues with altitude, you'll need to keep that in mind.
For hotels in Chamonix, I'd suggest checking the RS France guidebook if you have a copy, or the Trip Advisor listings at this website.....
https://www.tripadvisor.ca/Hotels-g187261-Chamonix_Haute_Savoie_Rhone_Alpes-Hotels.html
You can check directly with the hotels to find out if they'll be open at that time of year.