"Alacrity" means enthusiam or eager willingness.

The key phrase in the sentence is that the subject "never went to church," indicating he is not very religious.  This can rule out "zealot," "reformer," and "iconoclast," which, in the context of religion, all indicate an engagement with religion on some level. "Libertarian" can be ruled out as it describes a political position.  "Atheist," meaning someone who does not believe in God, is the correct answer.

The sentence discusses the "commander's odd statements and confusing instructions" as leading to the missing word being applied to him.  This means the correct answer will describe someone who does such things.  "Dotard," meaning an old and senile person, is the correct answer.

The sentence describes the "older brother" as being the cause of his younger siblings misbehavior.  This indicates the correct answer should mean someone who causes others to act or misbehave, the very definition of "instigator."

The key word in the sentence is "ironic." As it describes the missing word, then it makes clear that the correct answer will indicate that nicknames often are sarcastic or go against their perceived meaning.  "Misnomers," meaning incorrect names, is the best choice among the answers.

The sentence could be completed with a few of the answer choices, but the best choice must be found. The sentence notes the subject was both "an accused murderer" and that the missing word has been ongoing "after he was fully acquitted." "Vilification," meaning literally to make a villain of someone, is the best answer choice.

The sentence describes the doctors' inability "to stem the tide of the illness." This fact is paired with the missing word's description of what is happening with the "patient's diseased organs." "Atrophy," meaning to waste away and become useless, is the best choice among the answers.

The "companies' presidents" are described as discussing "their own labor practices."  This indicates the correct answer needs to have a meaning related to conspiring or working with each other secretly. "Collusion," meaning illegal plotting and planning in secret, is the best choice among the answers.

While all the answer choices can make sense in the sentence, one fits better than all the others. The sentence notes the professor's statements were "on female students," and all the choices have a negative connotation.  Of all the answer choices, "misogynist," meaning a hater of women, is the best fit.

The sentence notes the "young man" is concerned about "misdeeds and poor choices."  This indicates the correct answer must have a negative and even punishing connotation.  Only "perdition," meaning damnation and punishment, has the appropriate meaning and implication.