I would think you must apply to the M2 program as that's the one that leads to the PhD (also called the DEEQA - French abbreviation) program. I came from an undergrad and so I applied to the M1. The new applications can be made using the link you posted.
There is some information on grants and scholarships on the TSE website. The Eiffel Scholarship is a well-known grant, you could refer to previous posts on this forum for info on that. TSE has its own grants and akaik all students who get in to the PhD program at TSE get funded by TSE. So you need not worry on that. Just to be sure of anything, send an email to Aude Schoesling (email on TSE website).
Anything else you want to know?