would it be possible to play football manager on two computers at the same time if we were both logged into one steam account?

also after the game has been installed on the computer do you actually need to keep steam?

Both could login into steam in offline, although not quite sure if FM allows offline mode in steam (I know some games don't allow steam to play them offline).

Also if you choose to purchase or install your retail copy of FM via Steam then you need to keep Steam in order to play the game. Uninstalling it, would mean having used Steam in the first place as pointless.

Yes you can do it. Me and my bro do it all the time.

would we then be able to play a net game me vs him using the same bought copy from steam ?

No.