I don't think time counts in your score. Since this is an adaptive test, the number of questions you get wrong and their "distribution" is the important part. Experts can confirm that the time you have left at the end of your test does not influence your score: after all, you already have a time limit and this is not the math/verbal marathon!

Definitely not - how much time you take does not affect your score. Personally, I think it's the opposite - people who finish with 20 minutes left typically don't score very well! Personally, I always go down to the wire in quant and have a few minutes left in verbal. Unless you're scoring 99th percentile, finishing early is silly, since finishing early means you could've spent more time on questions you got wrong and, presumably, gotten more of them right.

It's not likely that the amount of time as you describe is a factor. Working at a steady pace is probably best. Rushing through can lead to careless errors.