11. Staring down his opponent, the bull had the upper hand in the fight.
Present participle, modifying "bull"
In this sentence the bull is being described as starting down his opponent.
12. I looked away when I realized I was staring.
Verb
The speaker is actually doing the action of staring.
13. Staring off into the distance, the smoke appeared to fade.
Present participle, modifying ? This is a dangling participle.
We don't know who is staring off into the distance...it's definitely not the smoke.
14. The car was flying down the road.
Verb
The car is doing the action of flying (metaphorically).
15. The car was going so fast it seemed to be flying.
Verb
The car is doing the action of going and flying.
16. Flying always makes me feel nervous.
Gerund.
Flying is the subject of the sentence, since it functions as a noun it is a gerund.
17. Running down the road, I felt better than ever.
Present participle, modifying I
Running down the road describes I.
18. Running always makes me feel energized.
Gerund
Running is the subject of the sentence, since it functions as a noun it is a gerund.
19. Running across the beach, sand was flying everywhere.
Present participle, modifying ? dangling participle
The sand is the subject of the sentence, but it is not running across the beach, so "Running across the beach" is a dangling participle.
20. Running across the beach sent sand flying everywhere.
Gerund
The action of running across the beach is the subject. It is what sent sand flying everywhere.