I've heard of the terms brush-up, release, stop, etc. When natives write, do they follow this rules? Is it widely known? Would natives know if I did a release, stop, etc? Is it okay not to use it as long as the character is identical to the original?
A well trained eye sees deviations even if we foreigners eyes think "it is identical"
Exposing one's hand writing to others is nowadays almost as rare in Japan as it is in many "Western" countries
As to 1: It is funny how our brains work also when interpreting what we see; we see some very subtle differences when we have learned to focus on them, while we really overlook others.
As far as I know, a person who has gone through the Japanese school system has seen the same characters so often and has has so many repetitive excerices of writing, that he/she is able to notice differences which we with less of that experience can't. (Compare that to "all the Japanese look alike when seen by foreigners" and "all the foreigners look alike" when seen by Japanese)
As to 2: (Even the use of fax machines in homes, is slowly but surely disappearing even in Japan...). In addition to not exposing one's handwriting, and thus not having to interpret the handwriting of the others, also for one's own use, the need of it is reducing. But, still, you write the way you have gotten used to / been trained. (Not only the Japanese, at least I write O,o and 0 counterclockwise because I am used to do that way, and, presumably this comes from school. (And the small alpha (α), while learning that a lot later, goes "clockwise" (ie starting from top-right)
How accurate your letters are doesn’t matter so much. It is widely known because there’s 書道(syodou) culture which is about how to write letter nicely. Japanese these days don’t care about how your letters look like if your letters are not hard to recognize.