The placement of the comma in this sentence between "then" and "a" is incorrect. The comma should instead appear between "vacation" and "but" in order to correctly connect the compound sentence's two independent clauses using a comma followed by a conjunction.

The comma between "loudly" and "when" is unnecessary, creating and punctuation error in the sentence.

The sentence requires a comma after "swimmer" in order to correctly join its dependent and independent clauses ("Although Danny was not the best swimmer" and "his friends always said he threw the best pool parties"). 

The sentence consists of two separate clauses ("While I don't agree with your opinion about how life on Earth began" and "I do respect your right to believe it"); therefore, the sentence requires a comma after "began" to properly separate the clauses. 

There is no error in this sentence. "Although" is a subordinate conjunction correctly linking an independent clause to a dependent one, "diligently" is an adverb correctly modifying a verb (studied), and "as she walked" correctly indicates an action that occurs simultaneously (to her fear and doubt).

Here, “before calling the police” is an interrupting or parenthetical (dependent) clause and can be removed from the sentence without making a sentence fragment. For this reason, it has to be set apart by commas. No comma is needed before the conjunction (“and”), since it is not breaking up two independent clauses. The sentence is correct as written.

Since “the marketing team spent all month coming up with an advertising campaign” and “the company chose a different and poorly conceived approach that the boss’s son had created on a whim” are independent clauses (i.e., complete sentences on their own), a comma must be used before the conjunction (“but”) that joins these two independent clauses. “In the end” is a parenthetical or dependent clause, so it must be separated from the rest of the sentence with commas on both sides (before “but” and before “end”).

Here, “Budapest, Austria-Hungary,” is a dependent clause that modifies or describes Houdini, so a comma is needed both before and after the clause. Because dependent clauses contain non-essential information (they can be removed and still leave a complete sentence), they must always be separated from the rest of the sentence by commas. Remember, too, that commas are always used between the name of a city and the name of its state or country, as in “Budapest, Austria-Hungary.”

In this sentence, “whose owners were disinterested socialites” is a dependent clause describing the racehorse. As with all dependent clauses, this one can be removed and still leave a grammatically complete sentence, so it must be set apart with commas on both sides.

“While it would be great if the Argentinian team won the championship” is a dependent clause, which means that it can be removed from the sentence and still leave a grammatically complete and logical sentence. Because commas are always used to separate a dependent clause from an independent clause, a comma must be placed between “championship” and “I would.”