Someone who is a “proxy” is one who represents someone else at a meeting or an assembly. In the world of computers a “proxy server” is one that sits between computer A and computer B, allowing A to (potentially) spoof B into thinking that the traffic is coming from somewhere else. For example, if computer P is the proxy, instead of communicating A → B and B → A, the communication is A → P, P → B, B → P, and (finally) P → A. Here, P is acting in “proxy” between A and B (like Meredith’s son).

Directly defined, a quagmire is a kind of bog that causes one to sink into its ground because of its softness. It is derived from the parts “quag” and “mire”—both words for bogs and marshes. In its extended sense, a quagmire is complex situation from which it is difficult to remove oneself. Here this meaning is indicated particularly by the adjective “complicated” as well as (to a lesser extend) “endless.”

The contrast communicated in this sentence is between those who understand things immediately while others must think about them quite a bit. Although such things might involve research, experiments, and the like, none of these are necessarily the case. (For instance, one could think over a math problem for some time, and none of the involved actions would fit any of those categories, strictly speaking). “Ratiocination” is a process of exact, reasoned thought, often implying a long train of necessary steps—i.e. proceeding in a non-intuitive (though valid) manner. The word is related to the word “rational” and other such “ratio-” words.

All that is implied is that certain words (“principle,” “cause,” and so forth) can take on various shades. It is fare to say that these would be different “modes” of expressing the same words. The word is often used in the sense of (e.g.) “mode of transportation.” Even in this case, you can see how the word means “manner of transportation,” implying a “given particular ‘shading’ of the general notion of ‘transportation.’”

The word “plebs” in Latin means common people or general citizens. A “plebiscite” is a full body of the people, called together to vote on a matter. This word is the best choice here because the clear contrast is between a representative vote and one that is non-representational or direct. The word “plebian” means one of the lower classes (i.e. of the “general run” of people in a society).

To be the “laureate” is to be one who has been publicly recognized for work (particularly of intellectual or cultural worth). The name comes from the ceremony of placing a wreathe on the head of the one so praised—which is related to the idea of the green leaves on such a wreath that would be made from laurel. The “laureateship” is merely the position of being the one so honored (as “kingship” is the status of being the king, etc).

Since information needed to be pried out of George and Robert, they would have to be questioned at length. Though we associate “inquisition” with events in the late Middle Ages and early Modernity in the Catholic Church, the term first means prolonged questioning. It is related to words like “inquire” or “query.”

The word “velleity” means really a passing, though not wholly motivating, wish. (It is like the attitude of saying, “I like it but can take or leave it”). It is related to words like “volitional” and “voluntary,” which are derived from Latin roots meaning “to will, wish, or want.”

The key word in the second sentence is “meaning.” The opposition implied is that many judge such questioners as doubting that there is any meaning to life at all. To be “nihilistic” means literally to believe in nothing. The word “nihil” in Latin means “nothing” and is found likewise in the word “annihilate.”

The key word here is “otherness.” The word “alterity” is not at all common, but with a bit of elimination and etymology, you can guess it. If you see that “otherness” is key, you will realize that really none of the other answers imply that state of affairs. (Indeed, many of them imply relationship and dependence—quite literally in one case). The “alter-” in “alterity” comes from the Latin meaning “other” or “another.” When you “alter” something, you make it to be “another.” Likewise, someone’s “alter ego” is their “other personality;” therefore, “alter+ity” is “the state of being other” (that is, distinguished from someone else).