First of all, congratulations for your success on having a position. As you have already realised, passing the selection procedures is only a step in the way of becoming a EU official. 

Being myself an AST/SC in the Commission, and not in the EP, I don't know if my own experience could have any value for you. You can always take it with a pinch of salt. But here we go anyway!

"What are the next steps?" in my case, I was called (with a lot of newbies) to a welcome day. We got our logins and passwords, and had some talks about practicalities.

"Do they help us to find a flat?" Not in the Commission. However, there is an internal list of flats to rent belonging to EC colleagues which is somewhat useful. I had also a revision of my rental lease, and got some tips in order to improve it.

"How long we have to wait for the first pay?" In my case, I started working on 1st August andwas paid , if I remember correctily, the following month.

I hope this will be of some help!