Let's go through your answers for each category:
Verb Form and Verb Tense:
8a. Since American Idol premiered on TV, many talented contestants have appeared on the show.
8b. Since American Idol premiered on TV, many talented contestants appeared on the show.
You chose 8b, and that is the correct answer. The verb tense in this sentence should be simple past tense because it refers to a specific event that happened in the past.
Identifying Antecedents:
10a. The antecedent in the following sentence is students: Students in online classes have to be organized to keep up with their assignments.
10b. The antecedent in the following sentence is class: Students in online classes have to be organized to keep up with their assignments.
You chose 10a, and that is incorrect. The antecedent in this sentence is "students." They are the ones who have to be organized. "Class" is not the antecedent because it doesn't match the pronoun "their."
11a. The antecedent in the following sentence is Doctors: Doctors should schedule more time for patients so they do not spend so much time in the waiting room.
11b. The antecedent in the following sentence is patients: Doctors should schedule more time for patients so they do not spend so much time in the waiting room.
You chose 11b, and that is the correct answer. In this sentence, "patients" is the antecedent because the pronoun "they" refers back to them.
Pronoun-Antecedent Agreement:
12a. When a driver wants to buy a new vehicle, he or she usually test drives them.
12b. When a driver wants to buy a new vehicle, he or she usually test drives it.
You chose 12a, and that is incorrect. The pronoun "them" is plural, but it should be singular to agree with the singular noun "vehicle." The correct answer is 12b.
13a. Most presenters spoke without looking at his or her notes.
13b. Most presenters spoke without looking at their notes.
You chose 13b, and that is the correct answer. The pronoun "their" is plural and agrees with the plural noun "notes."
14a. You may rent a movie or a video game for just one night if they are new.
14b. You may rent a movie or a video game for just one night if it is new.
You chose 14b, and that is the correct answer. The pronoun "it" is singular and agrees with the singular noun "game."
15a. Neither the mayor nor the council members have filed their reports.
15b. Neither the mayor nor the council members have filed his reports.
You chose 15a, and that is the correct answer. The pronoun "their" is plural and agrees with the plural noun "reports."
16a. Anyone who requests a copy of the game may have it for their video library.
16b. Anyone who requests a copy of the game may have it for his or her video library.
You chose 16b, and that is the correct answer. The pronoun "his or her" is singular and agrees with the singular noun "anyone."
17a. Each of the applicants must submit his or her completed form to the interviewer.
17b. Each of the applicants must submit their completed forms to the interviewer.
You chose 17b, and that is the correct answer. The pronoun "their" is plural and agrees with the plural noun "forms."
Pronoun Case:
18a. My brother knows more about technology than me.
18b. My brother knows more about technology than I.
You chose 18b, and that is the correct answer. The pronoun "I" is used after a verb like "knows."
19a. Please send an invitation to Bob and she.
19b. Please send an invitation to Bob and her.
You chose 19b, and that is the correct answer. The pronoun "her" is the correct objective form.
20a. My best friend and I are going to Hawaii this summer.
20b. My best friend and me are going to Hawaii this summer.
You chose 20b, and that is incorrect. The correct answer is 20a. When the pronoun is the subject of the sentence, it should be "I."
Pronoun Case for Who and Whom:
21a. Who was the star of last year's award-winning movie?
21b. Whom was the star of last year's award-winning movie?
You chose 21a, and that is the correct answer. "Who" is the correct pronoun to use in this sentence as it is the subject of the verb "was."
22a. To who should we send the letter of commendation?
22b. To whom should we send the letter of commendation?
You chose 22b, and that is the correct answer. "Whom" is the correct pronoun to use after the preposition "to."
23a. Jon Bon Jovi, who is a well-known singer and actor, also owns an arena football team.
23b. Jon Bon Jovi, whom is a well-known singer and actor, also owns an arena football team.
You chose 23b, and that is incorrect. The correct answer is 23a. In this sentence, "who" is the correct pronoun to use because it is the subject of the relative clause.
Vague Pronouns:
24a. When the car hit a tree, it made a terrible noise.
24b The car made a terrible noise as it crashed into a tree.
You chose 24a, and that is the correct answer. The pronoun "it" in this sentence is vague and doesn't clearly refer to "the car." 24b provides a clearer and more specific explanation of what made the noise.
Overall, you have many correct answers, but there are a few that need adjusting. Double-check your responses and make the necessary corrections.