Double-clutch, upshifting:
1. Put clutch in.
2. Shift into neutral.
3. Release clutch.
4. Let revs fall until theyll match the next gear.
5. Put clutch in.
6. Shift into appropriate gear.
7. Release clutch.
Double-clutching for downshifting is the same, except on step #4, you would blip the throttle to bring the revs
up to match the next gear.
Yes, but dont get hung up on the neutral part too much (as in, you dont have to wait until youre solidly into neutral before you blip the throttle you can do it anytime the clutch is disengaged).
Well, thats not exactly good for your clutch in the first place.
Your car mightve felt like it was going up and down because there was probably a backwards load transfer, which would then cause your front springs to expand and push you up slightly. I dont really know too much about that stuff though, so someone better-qualified should probably answer that.
Yes. Though Im not sure why thats so amazing.

Its very intuitive if its in gear, the momentum of the car is turning the axle, thus turning the engine, so no gas needs to be supplied to the engine. When its rolling in neutral, the axle isnt connect at all to the engine, so the engine needs to use a little bit of gas to keep itself moving (idling if you want to think of it that way) so that it doesnt stall.