Very pleased with the manager’s performance over the past two years, the boss gave the high-performing manager her bonus.

Correct sentence: Very pleased with the manager’s performance over the past two years, the boss gave the high-performing manager a bonus.

Using "a" helps to clarify the ownership of the bonus. This question is an example of a pronoun with more than one possible antecedent.

Both the car and the truck that crashed were so old that it would not be repaired; instead, both would simply be scrapped.

Correct sentence: Both the car and the truck that crashed were so old that they would not be repaired; instead, both would simply be scrapped.

From the second independent clause after the semicolon, we know that both the vehicles will not be repaired and would be scrapped instead. Therefore, "they" appropriately refers to both the car and the truck and makes logical sense in context of this sentence.

At the doctor’s office, she said that he had a liver disease that, while not life-threatening, should be cured as soon as possible with supplements.

Correct sentence: At the doctor’s office, the doctor said that he had a liver disease that, while not life-threatening, should be cured as soon as possible with supplements.

It is unclear, as originally written, who "she" and "he" refer to. "the doctor said that he" lets us know that the doctor told him that the latter had a liver disease.