To pick, do you rest your pick on one side of the string and move the pick so that it slide or falls off the pick, or do you move the pick so that it strikes the string and moves through it? And should the pick be affected by the string or should it just move unaffected by the string? And what muscle do you use? I can't move my wrist fast if my palm is facing the guitar. Should I rotate? Some people seem to be able to translate very fast but how?
Also, how do you move from string to string when alternate picking? It seems like you have to move in and out of the string with every pick.

Pick so that the pick moves through the string I'd guess. There are various picking techniques that can be applied. You can pick very lightly or drop the pick deeper into the strings depending on the type of tone you are looking for and you can also vary this depending on which part of the pick you use. You should really be picking from the wrist and there shouldn't be much movement from your arm really. The wrong way is to lock your arm up and pick from the elbow etc.

I pick so that the pick moves through the string. I use my wrist to pick and sometimes I just move my index finger and thumb backwards to pick.

I remember my first guitar teacher actually taught me to pick from the elbow and tighten up the arm... Luckily through watching instructional videos I learn that was the wrong way before I got started doing it :P

Do it the way is feels best for you and the music. But make sure you do it cleanly.

I'm not actually aware of how I hold my pick. I usually use my fingers and some wrist. When I pick, I anchor my pinky and ring finger after my High E string and put them on the body. I usually get a clean sound out of my guitar because the technique deletes any excess noise.