Trick-taking card game, where you aim to reach zero points?
This sounds like a trick taking game in the Rams group.


  Rams group: in these games players have the opportunity to drop out after seeing their hands, but anyone who stays in and fails to make their quota of tricks suffers a penalty.


None of these seem to match the rules you state exactly.

Raub starts at 21, with players trying to get to 0 points. But, it is played with 4 cards, when you fail to score a trick you add +4.

Loo: doesn't have the starting total, with players winning shares of the pot for the proportion of tricks they take, with players that take none having to pay into the pot.

Bourre: also isn't a perfect fit. It also has a pot like Loo, has players who fail to take tricks paying into the pot. It does have the first revealed card to indicate trump.