Colons can be used to set off a list. In this case the list is composed of three clauses. The sentence is correct as is. All of the other answer choices contain punctuation errors.

A colon can be correctly used to set up a definition. In this case, we are defining the definition which the subjects of the sentence pulled from the dictionary. Using an em dash is inappropriate. Em dashes are typically used to set up an interrupting clause that is not entirely relevant to the sentence. Using a semi-colon would also be inappropriate. The best answer choice simply uses a colon, rather than commas which result in punctuation errors. "Then" and "than" are commonly confused but "then," in this case, would be incorrect.

This sentence contains a semicolon where a colon should be. Note that semicolons only join independent clauses. The words "mats, oil, tools, and more" is  not an independent clause. Furthermore, these words describe a part of the preceding clause: what the subject bought. Colons combine two clauses when the second clause explains something about the first one. Additionally, the words "mats, oil, tools, and more" are in a list, so each item in the list should be separated by commas. The correct choice is, "his car: mats, oil, tools, and more."

This sentence contains a colon where one is not necessary. Colons are used to combine two related clauses when the second clause explains something about the first one. The colon used here, however, appears in the middle of a clause, which never happens with colons. The colon should be omitted and no additional punctuation is necessary. The answer is, "They liked to sing the songs."

This sentence misuses the colon. Colons are used to join two related clauses when the second clause explains something about the first clause. While clauses here are related, the second clause does not explain anything about the author writing many books. A semicolon would be more appropriate. The correct choice is, "She wrote many books; her most popular book."

The sentence has a cause and effect structure. The cramming all night led to him missing his alarm clock and oversleeping.  Because of the interconnected nature of the two parts, the best connector is a colon. 

In this case, the colon is the best option. The first part of the sentence introduces the following list of animals. Therefore, the colon is the best punctuation to separate the related parts of the sentence. Note that a colon can only follow an independent clause, which "These four are my favorite animals" is. Putting a period at the end of that independent clause, however, creates a fragment with the second clause.

The phrase "For the first nineteen years of his life" is an introductory phrase, explaining information necessary to the meaning of the sentence, but sitting outside the main construction of the sentence. Any such phrase needs to be set off from the rest of the sentence by a comma. Therefore, the only possible correct answer choice is "For the first nineteen years of his life, the young man."

In the sentence, the phrase "like art and gym" is an explanatory phrase, one that conditions the meaning of the sentence, but is not a part of the sentence's core structure. Any such phrase must be set apart from the main portion of the sentence with a comma. The only answer choice that correctly places in a comma is "which he found frivolous, like art and gym."

All of the commas in the original sentence are needless. The phrases "the play" and "our friend" are not significant enough (unless this is the last play in the world with the last friend in the world) to warrant appositive commas.