I started realizing that the picking hand was actually very important (probably as much as the fretting hand): it is always moving and it has to be extremely precise. I wanted to ask a question about some technique I saw in a video but I don't quite understand what's going on. In the video below starting at 1:57, the guitarist does some kind of "crunch/mute" sound in between the notes he's playing. I have a hard time figuring out how he's doing it. Is he doing that sound by using his right hand against the strings and then quickly lifting it off to play the notes or is he just doing that "crunch" mute sound with the left hand by just lifting his fingers off the strings between the notes? This seems to be related to his right hand doing a quick double movement (at 2:01), if someone could explain it, I would appreciate it because it seems that's how he's doing that sound.
This is mostly right hand muting (although I wouldn't be surprised if he also uses a quick left hand lift, as that helps kill the sound more quickly)
You have described it pretty accurately, he does quickly press his right hand against the strings between notes.
It's a very common thing to do, especially when using a lot of distortion and gain - it helps stop squeals and extra noise, and as he also has a compressor on this it helps drop the sound level quickly to give a sharper edge to the note. With practice it's almost automatic.
EDIT: what your updated question shows is actually the damping plus a quick pick to give a very short crunch/click noise. But the same process is true.