I now understand (at the end !!!) that something's wrong.
I actually don't see ANY files in the picker (except recently uploaded by myself, sometimes) - nothing at all. Our site admin told me that that was just like that in Moodle 2.2, but well, he's not right then. I've got admin rights (technically) but I'm not allowed to do any changes, I will contact him again.
Thank you for your explanations ! These will help me in the future too (I hope that the issue will be repared).