1. You place a tee in the ground to help mark the area where you intend to drop and then let the ball fall. The ball lands and stops in the drop area, but not before skimming the tee. Should you re-drop?
No. A small object like a tee or coin used to mark the drop area isn’t considered equipment
(definitions). If it were on the ground for another reason and the ball struck it, you would need to re-drop.

2. Watching your opponent take a drop, you see he holds the ball at shoulder height but flicks his wrist as he lets the ball fall, putting extra spin on it. If you call him out for doing that, can he correct the mistake without penalty?
Yes. If he doesn’t re-drop properly, it’s a one-shot penalty (decision 20-2a/2).

3. Before taking a drop, you grab another ball from your bag and drop it to see what the outcome might be when it’s time to do it for real. Is this allowed?
No.
It’s contrary to the purpose and spirit of the rules (rule 1-4). Penalty is loss of hole in matchplay or two shots in strokeplay (decision 20-2a/8).

4. A bush within the proper drop area catches your ball on the way down, and the ball gets stuck in its branches without ever touching the ground. Re-drop?
No.
The ball was in play when it struck the course within the proper area (decision 20-2b/1).

5. You drop a ball near the edge of a bunker, and it rolls in. Re-drop?
Yes,
even if the ball didn’t roll any farther than the prescribed distance, such as two club-lengths (rule 20-2c(i)).

6. As long as you extend your arm and let the ball fall from your hand, does it matter how high you position your arm?
Yes. Your arm needs to be at shoulder height (rule 20-2a).

7. Your opponent attempts to drop within two club-lengths of a lateral water hazard. The ball rolls into the hazard, requiring him to re-drop. Instead of doing that, can he choose another relief option – such as returning to the spot of his previous stroke?
No.
Once he dropped correctly under an applicable rule (26-1c), if a re-drop is required, the other options are no longer permitted (decision 20-2c/5).

8. A course enacted a Local Rule allowing for a drop zone on its island par 3. You choose to use the zone after plunking one in the water. You drop your ball, and it hits inside the zone but rolls away from the hole two inches outside the zone. Re-drop?
No.
The ball does not have to  rest in the zone (appendix 1-a-6).

9. You realise your ball will be lost forever in a water hazard after taking a drop outside the hazard but on the slope of its bank. Can you drop the ball and stop it from going into the water as long as you wait until after it crosses the margin of the hazard?
Yes (decision 20-2c/4). If a ball rolls to a place where a re-drop is required (e.g., a hazard), you can grab it before it stops, provided it’s likely it would not leave that area.