In the sentence as it is written, "fast," an adjective, is being used to modify "walks." This is incorrect; only adverbs can modify verbs. So, we need to pick out an answer choice which uses the adverbial form of "fast," which is "quickly." This lets us eliminate the answer choice "most fast." "Faster" cannot be correct because more than two subjects are being compared in the sentence, and "faster" is a comparative adjective that is used to compare two subjects. "More quick," "more fast," and "more quickly" can each be eliminated for this same reason, in addition to the fact that "more quick" isn't grammatically correct at all—one would say "quicker," not "more quick." This leaves us with the correct answer, "most quickly."

We must compare the chefs to my mother, not to the home-cooked meals.

The use of "more" in this sentence is odd and confusing, as "more" should be used in a direct comparison between two things. As no comparison is made in this sentence, the modifier should simply indicate a great degree of difficulty. The word "most," a superlative adjective, properly conveys such a meaning, making the correct answer, "The most difficult step in the process for all of the initiates was the lengthy obstacle course."

The use of "fuller" is incorrect in this sentence, as "fuller" implies a comparison, but is not comparing anything in the sentence. The use of the simple adjective "full" is perfectly appropriate in the sentence, making the correct answer choice "about the full extent of his condition."

Calories can be counted, and so we use the word "fewer." "Less" is for quantities that cannot be counted, such as water.  

As the question compares two people, a comparative form of the adjective in question must be used, meaning that "few," the regular form of the adjective, and "least" and "fewest," which are each superlative forms, are incorrect. While "fewer" and "less" might each look like a potentially correct answer, "fewer" is used to distinguish between countable nouns, while "less" is used to distinguish between uncountable nouns, like "water." (You can tell if a noun is countable or not if you could precede it with a number, and "seven water" doesn't make sense, given how "water" is used here.) So, "less" is the correct answer.

When comparing more than two items, you should use superlative adjectives. So, in this case, you should use the word "best."

The use of "they're" is incorrect in the sentence. "They're" is a contraction for "they are," while the sentence needs the third person plural possessive pronoun, "their." "Fewer people threw their support" is the correct answer choice.