The error in the sentence as it is written is that there it doesn't contain a verb. "Troubling" is actually a participle, a verbal that acts like an adjective; in the sentence given, it is describing "developments." To make a participle into a verb, one needs to introduce a form of the verb "to be" immediately before it; therefore, "In every part of the world there are troubling developments in politics" is the correct answer choice, as it is the only one that correctly inserts a verb.

The answer choice that changes "troubling" to "trouble" may look like a potentially correct answer, since "trouble" can function as a verb (as in "The questions he cannot answer trouble him"), but in this case, the verb "trouble" doesn't have a plural noun to refer to; it is followed by "developments," but "trouble" is a transitive verb—you don't just "trouble," you "trouble" something, so even if it were intended to refer to "developments," it would still be grammatically incorrect from having nothing to refer to.

The sentence is a compound one, featuring two distinct sentences, joined by a conjunction. As such, the second phrase, beginning with the underlined portion, needs to have a subject and a verb as with any sentence; therefore, "and they were able to take" is the best choice among the answers.

The underlined portion of the phrase is a dependent clause that is confusing because of the way it is worded. The phrase needs to be better linked to the main portion of the sentence. The answer choice which best does this is "and was called on when the force needed expert advice."

The fragment and the sentence can be joined together with a simple comma to make one complete sentence.

Because all three of these sentences can stand independently by themselves, separating the third with a semicolon keeps the new sentence from being a run-on.

Introducing the sentence with "even though" helps indicate that there is some contradiction between the two sentences.

The second sentence describes the dorm's location, so the following option is best because it allows the location as a description of the dorm:

"Maria was excited to move into her college dorm, which was conveniently located near her favorite campus quad." 

"Even though," "despite the fact that it," and "however" make it seem like the dorm's location is a negative detail, but the second sentence implies that this location is a positive detail through the use of the adverb "conveniently," so those options are incorrect.

The other option merely connects the second sentence to the first as a dependent clause that makes it sound as if Maria is "conveniently located near her favorite campus quad," so it is not as clear or grammatically sound as the "which was" option.

There are two grammatical errors in the underlined portion of the sentence. First, the two independent clauses need to be joined by the conjunction "and." Second, the two verb forms in the two clauses should be parallel, leaving "and wanted them to," as the only correct answer choice.

The answer choice "I should have stayed home from school today; because of this stomach virus, I am simply too sick to focus" combines two independent clauses with a semicolon, the second of which supports the first closely enough to merit the semicolon. 

Incorrect options:

"I should have stayed home from school today because of this stomach virus I am simply too sick to focus.": This answer choice is one sentence and a run-on.

"I should have stayed home from school today, because of this stomach virus, I am simply too sick to focus.": This answer choice combines two independent clauses with a comma, forming a run-on.

"I should have stayed home from school today, because of this stomach virus; and because I am simply too sick to focus.": This answer choice combines an independent and dependent clause with a semicolon, which you cannot do. The second part of the sentence is a fragment.

The underlined portion of the sentence contains an inappropriate use of the verb "overload." The use in the sentence needs to acknowledge the fact the final phrase is a fragment on its own. "Overloading" appropriately ties the phrase to the subject of "good ideas."