Can you use the word tournament for all sports?

Answer 1

Yes. A tournament is a competition with multiple competitors all competing in the same game or competition where a number of games must be played to establish the winner. (Thus an automobile race where a number of vehicles race at the same time is not a tournament; a situation where they race in a series of heats to establish a winner is a tournament). In some sports, the most common word for a tournament is something different--a curling tournament is called a bonspiel.

The word comes from "tourney", a medieval competition where pairs of knights would compete in turn.

Answer 2

Colloquially, yes--however, technically, a 'match' is an entire sequence of play, which may include games, sets, rubbers, etc.

The words "match" or "game" do not apply well to some sports where competitors compete by trying to get the best result from the same activity, including racing. In shot put for example people do not talk about "having a game of shotput". One factor may be that in such sports you may only try to outpace the other competitors and are not allowed to try to cause them to lose.