Is it considered an injustice for a coach to choose a competitor to represent the branch in a swimming competition without tryouts, and is it permissible for the competitor to benefit from the prize if he wins?

If your selection by the coach for the competition involves injustice and favoritism over someone more deserving than you, then do not participate in the competition. However, if the coach is authorized to select the contestants and deems you qualified, then there is no harm in that, and you may benefit from the prize if it is not from the participants' contributions.