Get into the bunker, play their shot then rake the bunker as they are leaving.
Nope
Tap the bottoms of your shoes with your club so the sand goes back in the bunker and then rake the bunker with the rake provided.
The Rake.
No, because this is viewed as testing, for which you would be given a penalty.
No, you are not allowed to do so, you must not rake the bunker or smooth sand over with your feet, as this is deemed to be improved your stance or testing the ground conditions. You will get a two shot penalty for testing, if you do so.
Yes, absolutely. Each bunker is considered an individual hazard, not a collective hazard of bunkers. If you play out of one bunker and land in another, you may rake the one you have just played from. Note, there is a common misconception that if you are in a bunker and rake footprints before you play your shot you are penalised, this is not the case, you are only penalised if you improve the lie of the ball, your stance or test ground conditions.
Yes, you are allowed to do this. It is one of the biggest misconceptions in golf. You are not allowed to improve your stance or lie, so if the footprints are on the other side of the bunker it is fine to rake them. However, since most people are unsure of this rule you may want to leave it until you play your shot as to not raise any questions. As I said, it is perfectly legal to do what your question asked.
Tap the bottoms of your shoes with your club so the sand goes back in the bunker and then rake the bunker with the rake provided.
Yes, you may do this, but set it down gently and out of the way, just in case someone accuses you of testing the surface.
Generally, no. Loose impediment. 2 stroke penalty. If that stick is attached to another stick with a bunch of little sticks (i.e., a rake for the bunker), then yes. Also, you cannot remove leaves from the bunker or loose pebbles. having said that, don't ruin your wedge grooves by playing out of crappy bunker conditions if you're just practicing. Pebbles reek havoc on forged wedges.
When you play a shot from a bunker and hit it out of bounds, you clearly have to replay your shot. But the rules stipulate that the conditions of the previous shot must be recreated. So if you played from sand which was raked, you would rake the sand and place the ball as close to where the last shot was played from in the last shot. If you were in a footprint you would have to make a footprint and place your ball in it, and if you were in a divot, you would make a divot and place the ball in it.