Yes, it is possible to solve puzzles in under a minute on the first try. On a lot of the puzzles in the recent months, I've been only attempting them once, and seeing how good of times I can get. Thought the main reason is I just don't want to spend a lot of time in my day playing sudoku. The ones recently that I've attempted more than once were usually not on the same day, but after I've seen that my first time was not that good. Then I go back and try harder on that one.
And in some of the earlier puzzles, people getting really fast times by multiple attempts (myself included), doesn't have to be by writing down the answer and cheating. By just doing a puzzle more than once successively, you start to remember what numbers you looked for first, or what areas you did first the last time. That speeds it up greatly. Is that still a form of cheating? Probably. I think the best scoring system would probably be no two attempts within 12 or 24 hours of each other, or something like that.
And as regards to the 0 second times, I've gotten a couple of those. It was times that my browser (firefox) froze after loading, so the timer never stopped, but I could do the puzzle. It records the time as 0 upon completion, and then my browser can't go to another page, so I have to close and open it again. Maybe Adam can put something in to automatically discount 0 second times.
And it's not hard to type super fast for these. I just always have my right hand on my mouse, and my left hand on the number keypad at the right side of the keyboard. If you've ever done any computations, you should have that keypad memorized. Then you never have to look at the keyboard.