This is a frustration for me. I have trouble understanding why I can finish a puzzle before time goes into minus and still end up with a negative score. Is the clock simply there to give you an estimate of the puzzle's difficulty? If not, wouldn't it make sense to make it so that the countdown and the points both cross zero at the same time? If you get it right, after time runs out, you should lose points. If you get it wrong after time runs out, you should lose more points. But if you get it right within time, you should get something. I'm sure there's complicated math behind it, but this doesn't seem illogical to me. Is it frustrating to anyone else?

I agree.  What I would like is time to look at the board before the computer makes the move so I don't have to analyze the whole board plus what the computer just did.  I think a score-bar like Chess Mentor has would be more helpful.  The clock seems to be there just to annoy you ;-)  ...I don't think it's a barometer of difficulty.  The 'countdown' idea would add pressure.  I'd like to remove the clock entirely or until I've solved the puzzle.

Take a look on the help page for Tactics Trainer. That gives a general idea of how it works.

The trainer is based on the idea that a person of a certain rating should be able to solve a tactic in a certain amount of time, based on the average time it takes to solve the problem. If it takes a player a significantly longer time to solve, then their tactics rating is not accurate and you lose point so reflect that.

The tactic is treated just like an opponent in a game of chess. If you run out of time, you lose the point and your rating decreases, just as it would in a game of chess.

Clock is 2x average speed, if you take longer you only get 20% credit. The rating change depends on your rating vs puzzle difficulty, so you can't have a time cutoff point for +/- very easily.