Pressed for time but a short answer may still be worth a little: often good players will stall out a little, their hips go forward, their arms leave their torso, and they have problems hitting big hooks, pushes, even push-cuts if they keep the face held open (they get tired of hitting big hooks) too much. So feeling that they keep their turning rates up can help eliminate those problems, keep the club exiting on plane rather than high, and keep their speed through the ball up so they don't have to be as handsy/wristy with it (you said something about your release being quieter).
Hip slide (and a little of the "jumping" that we talk about - extension of the front knee) is the transfer of weight/force/pressure. They're the same.
Hip rotation happens simultaneously with the hip slide. It changes the dimension of the hip slide - from 45° to the right at the top of the backswing (forward is towards first base), to 45° left just after impact ("forward" is towards third base).
The right foot doesn't push off, no. I answered that in the hip slide thread briefly.