The part after the semicolon describes what should fit into the first blank. Two researchers have differing opinions, this leads us to believe the first blank will mean confusing or ambiguous, perhaps even doubtful. For the second blank, we have a shift where the findings are described as "compelling." Look for words that mean the opposite of compelling. “Equivocal . . . conjectural” combine to form the correct answer.

The doctor’s methods were indeed interesting, as the sentence states. The fact that they are both new and questionable at least hints that they believed his methods to be a bit outside of the bounds of common practice. Whenever something is “orthodox,” it is part of the overall accepted scheme of a given discipline or social institution. Often, we think of the contrast of “orthodoxy” and “heresy” in religious matters. There can, however, be “orthodoxies” in many fields—things that are taken for granted as part of the status quo. 

Now, it makes sense to call the doctor’s methods (as described, at least) “unorthodox.” The implication is that they are either not dangerous or, perhaps, not illegal. The second meaning is what works for the second blank, for “illicit” means against the law. Something that is “licit” is something lawful. The sentence states that his methods were not illicit, even if they did seem to be questionable.

The first blank for this sentence clearly needs to be something negative. In the second independent clause (after the "for"), it is indicated that the action was, in fact, neither intentional nor very harmful. The sentence is basically saying that if you were to say it was some kind of harmful action, it was only harmful in a limited sense. The word "malicious" is the only one that is negative in this manner. True, something that is "tragic" is also negative, but we usually talk about malicious actions as being done by someone, whereas tragic things happen to someone.  To say that something is "attenuated" means that it holds in a limited sense or manner. This word is related to "tenuous" meaning weak (and thin), as in the expression, "The death was connected only in a very tenuous manner to the injury."

This sentence focuses on Mark's ability to speak. This should give you a preference for options that pertain to speaking skills. Eloquence is just such a word, describing someone who can speak very well. (It is related to numerous other words that derive from the Latin word for to speak, "loqui." Related derivatives are "locution," "elocution," and "colloquial.") The word "prodigious" means to a great degree or of a very large extent. This seems to describe well the large crowds that are hinted to have gathered for Mark's speeches.

The best clue for the first blank in this sentence is the fact that the scholar seems to specialize in the history of the Catholic church—at least as far as we can tell from this sentence. Now, this might lead you to pick "religious" or "pious" since the Catholic church is a religious institution, but what is is more proper is to say that the historian is an "ecclesiastical" historian. This literally means that she was a church historian. ("Ecclesiastical" comes from the Latin ecclesia, meaning gathering or, in later medieval Latin, church.) This is a much closer description of her role. The apparent point is that she had to research non-ecclesiastical matters. The best option for this is "secular," which means of the world. Another option would have been "profane," though that word really is the opposite of holy.

The easier blank to fill is the second one. Something that makes people fall asleep is a "soporific" agent. This word comes from the combination of the Latin root "sopor," meaning deep sleep, and the "-fy" suffix that means to do or to make. Something that is "stupefying" is something that makes it difficult to be fully conscious. It is not so much about being "stupid" as it is about not being in control of one's senses.

The most direct clue in this sentence is the word "self-gratification." Apparently, Laura's husband believes that she is only buying chocolate for the sake of her own selfish desires. Likely, he thinks that she is spending too much on chocolates. The only word that matches well, therefore, is "exorbitant," which describes too large of an amount of money. Something that is decadent is self-indulgent, so the word "decadent" works well for the first blank, as Laura apparently did not think that her preferences were self-indulgent, even if her husband did think so.

The first blank is clearly a contrast to the word "humbler."  We do not know if the young man was arrogant in his goals. That kind of negative character judgment is not indicated by the sentence's clues. It is fair to say that he was bold in his hopes. Thus, the word "audacious" is a good option, for it indicates someone who is willing to undertake bold risks. It seems that he also intended to stick to his convictions and intentions. Thus, he was "resolute" in them. (Think of this as meaning that he had firmly resolved to fulfill them.)

For these two blanks, we are looking for a pair of synonyms. This pair of words should be antonyms to "benevolently." The word "benevolently," as well as the related "benevolent," come from two roots with which you are certainly familiar. The introductory "bene-" comes from the Latin for well (as in "well done"). It is found in words like "benefit" and "benediction." The "-volent" portion comes from voluntas, meaning "the will." Thus, to be benevolent is to be good willed. Thus, we are looking for a pair of words that generally mean "unkind." The word "vituperative" means "insulting or abusive."   Scathing means very critical (often implying that it is quite nasty).

The second blank in this sentence is probably the easier of the two blanks to figure out. Then, after filling it in, you can guess the first blank rather easily. The second blank clearly is something like "sturdy." Thus, the word "robust" works very well as an option. This is far better than "weighty" and "broad," for neither of these words necessarily indicate sturdiness. Now, the first blank is the opposite of "sturdy." For this, "frangible" is excellent. It means able to be broken. It comes from the Latin frangere, meaning to break. This same root is related to the English word "fracture."