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. 

"The GMAT builds character" and "it is full of challenging sentences" are independent clauses. They are related, but independent; therefore, the best way of separating them within the same sentence is with a semicolon rather than a comma. 

The comma inserted in the underlined portion of the sentence between "margins" and "over" is unnecessary. The full phrase should run continuously without interruption, as it is all one thought. The only answer choice with appropriate punctuation is "in profit margins over the final quarter of the year."

The phrase "All his life" is an example of a introductory phrase, explaining information necessary to the context of the sentence, but outside of the main sentence, grammatically. These types of phrases must be followed by a comma in order to separate them from the rest of the sentence. The correct answer choice is "All his life, the boy wanted."

The phrase "a convertible" is a paranthetical expression: a phrase that explains more about a particular word or phrase in the sentence, but is not introduced by a relative pronoun. These phrases must be introduced and completed with a comma in order to separate them from the main clause of the sentence. The correct answer choice is "wanted, a convertible, for her 30th birthday."

The phrase "When he picked up his kids from school" is an example of a introductory phrase, explaining information necessary to the context of the sentence, but outside of the main sentence, grammatically. These types of phrases must be followed by a comma in order to separate them from the rest of the sentence. The correct answer choice is "When he picked up his kids from school, he surprised them."

The words "big" and "blue" are coordinate adjectives: adjectives that modify the same noun separately, but are not separated with a conjunction. Coordinate adjectives need to be separated with a comma. The correct answer choice is "They wanted a big, blue banner."