Something else you may notice is a lot of players seldom use open a, d, or g (standard tuning for simplicity) unless absolutely necessary. Once again, for reasons of crispness to the note, and also for moving to the next note. One thing I was shown when I first started playing 18 years ago was to avoid using an open string if at all possible. I didn't "get" it at the time, but several years later it became more and more apparent as I diversified what I was learning/playing/writing.
Also, there is no such thing as "correct" fret hand posture. The correct way is the way that is comfortable for the player. If you have to hang your thumb over, then do so (hell, use it to fret a note if you want!). That little bit of fret buzz is just fine, and also gives a great indicator of a smooth consistent note release timing. I like it, and really makes me feel OK with my playing as I try not to avoid little buzzes like that at all anymore.