It is always helpful to try to fill in the blanks for a sentence before choosing the answer that best fits. For this sentence, you will likely choose "show off" for the first blank and "informed" for the second. Whenever we flaunt something, we show it off in a very open manner. This is the only option that really expresses this idea of "showing off."  For the second blank, "erudite" works well, for this word means very knowledgeable. Clearly, the student wants to seem much more knowledgeable than his supposedly uneducated relatives.

Although we might find such stealing to be morally repugnant, the sentence really is not focusing on that. It is just saying that it is a pretty simple and easy achievement for one to steal candy from a baby. Whenever something is "facile," it is easy in this manner.  The word comes from the same roots as "factory," "facility," and "fact." All of them have something to do with doing or making. Something that is "facile" is easily done or made. These things require little effort; hence, they require little exertion.

Brothels are quite negative places of prostitution, which would likely give a section of town a reputation for being seedy or disreputable. It is implied, however, that the character of this section of town was improved much over the years. The word "ameliorated" means improved or made better. It comes from the Latin "melior," which means better. This is closest to the immediate description of the area made in the closing clause of the sentence.

It seems that the family in question had "made up" the story about the person's intelligence. Whenever we "fabricate" something, we do such making up. Of course, the word "fabricate" can merely refer to physical making. The word often has the negative connotation of making up something for the purposes of deceiving others. The word "vaunted" describes something that has been praised, generally too much. This certainly seems to be the case with regard to the man's intelligence, given how his family had basically fabricated the whole attribution of such intelligence to him.

The disinterest of the students is the best clue for assessing this situation. Likely, they are staring back at the professor with empty and disengaged stares. These could be called "indolent," indicating a kind of laziness. It would be better to look for something that more directly indicates the lack of interest. The word "passive" works well in this case, as it helps us to imagine the students staring back without really any external reaction on their faces whatsoever. The professor seems to want them merely to talk about anything whatsoever. That means that she would like them to engage her in a conversation on any topic. To be engaged does not only have to deal with getting married!  It can merely mean "to be involved with in some way."

The idea is that Peter has a kind of unspoken bias for drinking dark coffee. He does not even think that he has this bias. A predilection is a kind of inclination or preference for something. It actually comes from roots meaning to choose before. The "-dilection" portion of the word is related to the word "election" in English, as well as to "delight."  Both of these words have to deal with choosing or selecting something. Peter's predilection evidenced itself in his rarely choosing to have coffee that was not very dark.  To "opt" for something is to choose that thing. (Clearly, this is related to "option", which indicates a thing that is—or at least can be—chosen.)

The idea is that Byron's wealth helped to increase his social standing. The word "aggrandize" means "to make larger." Clearly, it has the root "-grand" in it.  The prefix "ag-" is a form of "ad-", meaning "to or toward." To "aggrandize" is to make something "go toward" being larger (or, grander) than it already is.  

The wealth itself seems to be endless. It is appropriate to describe it as being seemingly inexhaustible. To exhaust something is to use up that thing. Hence, we use the word "exhausted" to describe the physical state of being tired after exertion. The word is used in the sense of meaning having used up all of one's energy.

The implication of the sentence is that Todd's hope increased as the wallpaper finally came off the wall. Whenever we bolster something, we support it. The word can be used in a physical sense, as when something helps to support part of a physical structure. It can also be used in an extended sense to speak of any such "supporting" or "strengthening"—as when Todd's hope was bolstered by the favorable turn of events. 

As regards the second blank, it seems that Todd had thought the task was  impossible. This would be an "insuperable" task. This word uses the "in-" prefix in a privative sense, meaning not. Something that is "insuperable" is something that cannot be overcome.

The first blank expresses a contrast to the action of waiting. Instead of waiting for the mice to enter his house, Thomas is going to take preventive actions. It is better to choose a word like "preventive" than options like "aggressively" or "forcefully", as we really do not have evidence of such implied force.  The closest wrong option is "defensively", but "preventively" expresses the idea taking an active role in a slightly clearer manner.  

Likewise, with regard to the snares, we really don't know if Thomas is going to slay (etc.) the mice. Really, all that indicated is that he plans to trap them. Thus, "snare" is the best word, as it merely means "to trap."

The implication of this sentence is that Lee general was other-focused. He needed to at some point pay attention to his own concerns. Therefore, the best description for him is "altruistic." The prefix "atlr-" in this word is related to "alter," meaning other or another. To "attend to" something is to pay attention to that thing. Based on the remarks above, this is clearly what was necessary for Lee to do.