Do rugby clubs do transfers like in football? Do teams buy/sell players for a fee or do the players have to run out of contract or be released to move?

It depends on the nature of the individual contract between the club and player, but in general yes. Peter Stringer, for example, was "rented" by Munster to several English clubs last season. Munster didn't have a regular spot for him to play at the top level, but he is (or at least was) too useful a player to release outright, and they wanted to keep him sharp. The extra income he was able to generate didn't hurt either. This, of course, applies only to Northern Hemisphere clubs. In the SANZAR countries contracts for the top level players are with the national Unions rather than the individual clubs. It changes the entire nature of the legal relationship.

I don't know