This sentence features an error of subject-verb agreement. The subject of this sentence is singular: "the recent discovery," though the following elaboration of short films and a personal diary can be misleading they do not change the grammatical subject of the sentence. Therefore the verb should be "allows."

"Looking down" is the correct answer. 

The subject of the main clause (after the comma) is "faces" (particularly "Stacy's friends' faces"), which cannot logically be the subject of the first clause. This disagreement creates a dangling clause. It must be the subject (of the whole sentence) who is "looking down," but one can infer that it is Stacy who is at the summit, because her "friends' faces" "smiled up at her."

In this sentence, the pronoun “it” does not agree in number with its antecedent “efforts.” The corrected sentence reads: While I appreciate your efforts to ensure that the event runs smoothly, they are interfering with my ability to give the artists the flexibility they require.

This sentence has a problem with subject-verb agreement. The verb “makes” does not agree with the plural subject “noises.” The corrected sentence reads: The buzzing noises emitting from the alarm clock make me cringe every time that I hear them.

This sentence does not use the correct form of the verb that follows “found.” The corrected sentence reads: Harold had always found the specialized jargon of artisanal coffee shops to be confusing and intimidating.

This sentence uses a clunky progressive tense when it is not needed. The corrected sentence reads: His family grew up eating a steady diet of meat and potatoes, and now they are reluctant to try new foods like enchiladas and kale chips.

The correct answer is "they."

This is a case of subject-pronoun disagreement. Although the subject in the first clause is "the budget," the subject becomes "the Board of Directors" in the second clause. Therefore, the pronoun must be replacing that subject, and the subject is singular ("the Board of Directors"), while "they" is plural, so it is a definite error. While there are several people, at least more than one, in a Board of Directors, the subject is not the directors, it is the entity itself— the Board. There is only one Board.

Though "was not decided" is in the passive voice, it is not grammatically wrong, just weaker writing. "By" is the correct usage, "and had"  is a verb-verb agreement, both are in past tense.

"Between Jenny and I" is incorrect. In this case, the sentence should say, "Lauren sulked all during school because she felt 'left out,' but the conversation was between Jenny and me, and I had every right to keep it that way." This sentence has a pronoun case error. Since Jenny and the speaker are objects (grammatically) within the sentence, the objective case pronoun ("me") is required, instead of the subjective case pronoun ("I"). In order to better understand these situations, when the phrase includes a noun and a pronoun simply remove the noun before the pronoun, for example, remove "Jenny and." Once you do that, you will much more easily identify the correct pronoun.

Another example: "Dad bought Roger and I a bicycle." As opposed to: "Dad bought (Roger and) I a bicycle." One would not say: "Dad bought I a bicycle."

In this sentence, the pronoun “it” does not match in number with its antecedent “Members.” The corrected sentence reads: Members of the blue team travel around the city collecting buttons, and they will not be able to communicate with the red team at any time.

The error here is "most thorough." Since "thorough" is followed by a verb it is an adverb, and thus must become "thoroughly." The correct way to write this would be "the most thoroughly researched." Adding the "-ly" to an adjective usually makes it an adverb, and every time and adjective is describing another adjective or a verb it must become an adverb. The "it" at the end is the correct pronoun, as "it" is referring to the paper.