The phrase "Under the old rules" is an introductory phrase, one that conditions the meaning of the sentence but remains apart from its structure. All introductory phrases must be set off by a comma. The only answer choice that inserts such a comma is "Under the old rules, that kind of shot was perfectly legal."

In the sentence, the phrase "a mercenary from a foreign country" is an interrupting phrase, one that provides information but is not critical to the structure of the sentence. All such phrases need to be set apart from the sentence by a comma; therefore, the correct answer is "The soldier, a mercenary from a foreign country, was."

The phrase "tired from their long march" is a non-restrictive interrupting phrase, a phrase the provides information, but is not necessary in the sentence. Such phrases always must be set off by commas. Therefore, "The soldiers, tired from their long march, gladly settled" is the correct answer choice.

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 underlined portion of the sentence contains an interrupting phrase, "prone to overly assertive maneuvers." All interrupting phrases must be set apart by commas from the rest of the sentence. Therefore, the correct answer is, "The general, prone to overly assertive maneuvers,."

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 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."

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 comma in the sentence is incorrectly deployed, as it divides a continuous phrase into two parts. The sentence should actually have no commas in it, as the sentence works as a continual phrase. Therefore, the correct answer choice is "The hardest thing for most new pianists to lean is the correct hand position."

The phrase in quotation marks, "What goes around comes around," is the subject of the sentence, but described as an idiom. This means that it needs to be treated as a full sentence, because it is being referred to as a singular whole. To be a complete sentence, a comma needs to be put between its two parts, making ""What goes around, comes around" is an idiom," the correct answer choice.