Hi @ShinySheep, welcome to OGS and to the forum.
Tournament games start automatically when the tournament or a new round of the tournament starts. You cannot start the games on your own.
The number of games per round depends on the tournament type.
The only active tournament you are participating is a “Double Elimination” tournament. This type of tournament has only 1 game per round for each player. If all games of the first round ended, the next round starts with 1 game for each player. If a player loses 2 games in this tournament type, they don’t take part in the next round. The tournament takes as many rounds as needed to eliminate all but 1 player.
The other tournament you joined is of the “Round Robin” type. It’s a single round everyone vs. everyone tournament, so there will be 1 game for each other participant waiting for you when the tournament starts.