If you reinstall steam you have to rinstall the games. Make sure your view is 'all games' so you can choose what games to install.

TF2 progression will be the same as well.

Make sure you'r elooing into the correct steam account

Thanks for your answer !
But I'm logged yet and when I launch TF2 it's like i was a brand new player... Is all my stuff lost ?

I can't see your profile because its private, but as i said tf2 progress and achievments and inventory are stored on the cloud. reinstalling steam doesn't do anything to that. again be 100% sure you're logging intot he 'correct' profile

Are you logged in to the same account?

Ok, that's probably the problem. I tought that this kind of files were stored on the cloud too. But i simply don't see where i should login...

In the client, click on Change User... and then enter in your correct account details.

But what i don't understand is that Steam recognized my account yet, with the fact my real name, my mail and my pseudo are correct. So i still don't really understand... :/

You will have created a new account instead of logging in to your old one. It knows your name because you need to give it your name when you create a new account.

if your profile wasn't private we could see your join date etc. and game time played so we could have an idea of if this is true or not.

On a fresh OS install (drive formatted, etc) then steam, then TF2 I sometimes don't see all my info. However, the second time I start TF2 its all there...

So start TF2 a second time. If its still missing info (your inventory, achevements, etc) then its connecting to an account that doesn't have it. Go to the properties of TF2 on the main steam window and look for the "use Steam Cloud" checkbox, make sure its checked and then start again.

Sorry guys, was my bad : I effectively didn't log in correctly. But Steam let me believe i was in the right way...
Thanks for your answers.