"Under the teacher's guidance" is an introductory phrase, one that conditions the action of the sentence, but is apart from its main structure. Any introductory phrase needs to be set apart fom the main body of the sentence by a comma. The only answer choice that correctly uses a comma to separate the phrase is "Under the teacher's guidance, the students' reading ability improved."

The phrase "More than anyone else he had encountered" is an introductory clause, a clause that conditions the information provided in the main body of the sentence. All such clauses need to be set off from the rest of the sentence by a comma, making "More than anyone else he had encountered, the preacher was convinced of the rightness of his cause."

The phrase "Under the old regime" is what is called a conditional phrase, as it gives the conditions which make the main body of the sentence possible. Such phrases always need to be set off from the main part of the sentence by a comma; therefore, "Under the old regime, the action was," is the correct answer choice.

The phrase "a respected team leader" is what is known as an interrupting phrase, or a phrase that provides information but is unnecessary for the structure of the sentence. All interrupting phrases must be set apart from the rest of the sentence by commas; therefore, the correct answer is "The athlete, a respected team leader, was."

The phrase "long time veterans of the company" is an interrupting phrase, one that adds information to the sentence but remains outside of its main structure. An interrupting phrase must be set apart from the rest of the sentence by commas. The only answer choice that correctly sets apart the phrase is "The entire staff, long time veterans of the company, was surprised by their CEO's sudden resignation.

The phrase "living in foster care for years" is an interrupting phrase, one that adds extra information to the sentence, but is outside its main structure. An interrupting phrase must always be set apart from the sentence by commas. The only answer choice that appropriately sets off the interrupting phrase is "The young girl, living in foster care for years, had a tough life throughout her schooling."

The phrase "a shrewd negotiator" is an interrupting phrase, a phrase that has no bearing on the construction of the sentence but provides additional information. All interrupting phrases must be set apart by commas; therefore, the correct answer choice is "The executive, a shrewd negotiator, always."

The issue with the sentence is that its single word introductory clause, "Incidentally," is not appropriately set off from the rest of the sentence with a comma. The only answer choice that does that is "Incidentally, the boy was never at the site of the crime he was accused of committing." All other answer choices are grammatically incorrect.

The phrase "having never failed to get a date for one of her clients" is an extremely long interrupting phrase, a phrase that adds information to a sentence but remains outside the main structure of the sentence. All interrupting phrases must be set apart from the rest of the sentence by commas. The only answer choice that appropriately does this is "The matchmaker, having never failed to get a date for one of her clients, worked desperately before her deadline from her new client."

The phrase "anxious about their grades" is an interrupting phrase, or one that provides extra information, but is not crucial to the structure of the sentence. All interrupting phrases need to be set apart from the rest of the sentence by commas. The only answer choice that does this is "The students, anxious about their grades, were."