Do I need a free game slot for me to take part in a tournament?

You are assigned game slots and tournament slots which are separate entities.  So, no, you don't need to keep a game slot open to join tournaments, but, your tournament slots are also limited to ten as well so you'll need to manage them too!

Thanks. Now I feel bad not joining all those tournaments because my slots were full :-(

If you play many round robin tournaments you should consider adjusting your tournament settings to increase your "multiple tournament rounds" to somewhere from 5 to unlimited, or whatever you are comfortable with?  To get there click:

settings -> scroll down to "tournament preferences" -> click on the pull down for "multiple tournament rounds" -> choose a number of games you are comfortable playing in any one tournament at one time.

The more people in each tournament that have increased this preference the quicker tournaments finish and the more quickly you can join another!

The default is 1, but, I've often felt that Tom should consider changing the default to 3-5 or something to counteract non-active players joining tournaments then letting all their games time out one by one.  If the default were upped then they could at least time out of games three by three or five by five and not hold up all the honest players who are here to play!