Does FIDE make any money from media coverage? Unlike any other professional sports I know of, everyone seems to be able to broadcast chess world chess championship online. 

That raises the question, are they making any money from media coverage? Can't they sell exclusive coverage to selected media?
FIDE have sold the rights for the current world championship cycle to Agon. Given the very high levels of dissatisfaction with Agon it is unlikely that the contract will be renewed and the rights will revert to FIDE. There has been discussion of this on chess.com here.

Note that the moves of the game cannot be legally protected via copyright laws etc. Only the live pictures of the players playing the games can be protected like that. This has already been tested in the courts in several countries including the US (the most lucrative market) by Agon for previous matches.