Hi Mathias! Welcome to the forums.
The examples in the book aren’t comprehensive, but my reading is that Checks expire at the end of the Player Turn, but the Player Turn doesn’t end until everyone has had a chance to spend their checks.
When a player doesn’t make a test (either because they can’t or choose not to), they get removed from the Player Turn order and lose any remaining checks they had; since they get skipped in future passes, they also can’t be passed checks by other players to spend for new tests.
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No, players will never carry checks over from the Player Turn into the GM Turn. However, you won’t have more than 1 player losing checks at the end of the Player Turn since a player declining to test removes them (and their checks) from the player order but does not end the full Player Turn unless no one else is able to take a test (free or with checks). That means that no player will be forced to discard checks by another player declining to spend their own.
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You can choose not to spend your checks on your pass in the Player Turn, but declining to spend a check or take a free test will immediately cause you lose any remaining checks and be skipped in the next round of testing.