Although we tend to think of “proselytes” as persons who preach their belief (which they often do), the term most particularly indicates a recent convert.  Since the sentence states that this person has a “newfound” ideology, this is the best option to choose. 

Although a number of these options are likely tempting, note both the mention of “affectionately” and “informal,” indicating that the necessary word should reflect these facts.  A “sobriquet” is a nickname (sometimes descriptive—as is “lefty”), so this word best meets the needs of the sentence.

Fundamentally, “asylum” means “refuge,” a meaning derived from a similar Greek root.  Normally, people think of “asylum” in the sense of a “mental asylum,” but even that term implies such “refuge,” though in that case it is a refuge for persons suffering form mental illness.  Often, the term is used to imply that someone is seeking refuge from political persecution.  Since the sentence speaks of the political ills and persecution in the family’s native land, “asylum” is the best choice.

Here, you must be careful not to imply anything in the sentence.  Nothing is said about the state of the two person’s friendship.  Therefore either “amity” / “friendliness” or “quarreling” / “bickering” would imply that something has changed with regard to the relationship between the two persons.  However, since we have the word “cohabitation,” it would be best to choose this option, as it merely implies that the two friends had been living together.  The word comes with from the root “co-” meaning together (as found in “cooperate” and “correlate”)  combined with “-habitate,” which is derived from Latin roots meaning “to live / inhabit.”  The later has many related English derivatives such as “habitat,” “habitual,” and “inhabitant.”

This sentence might be difficult because of the temptation to ignore the word “diet.”  In actuality, this word is the most specific and matching for our purposes.  The word is used to name the legislative assemblies of some countries and hence is a better option than “group” or “congregation,” which do not match as well.   For example, in the city of Worms in 1521, an assembly gathered to address the questions related to Martin Luther.   This formal gathering is famously called the “Diet of Worms.”  As you can see, it has nothing to do with strange eating habits.  It merely means that an assembly within the Holy Roman Empire happened to convene at a city named “Worms.”

Although we often think of “orthodoxy” as being associated with religious beliefs, it is also applicable to any theory or doctrine, meaning that it is authoritative and generally held.  This word is the best option, given the adjective “unquestioned,” which does imply that the needed word have sense of implying a certain authority.  The word comes from the combination of “ortho,” meaning “straight” or “correct” and “doxos” meaning “opinion” (or “praise.”)  An “orthodontist” works to make teeth (“-dontist”) be straight (“ortho-”).  A “paradox” is something that seems to be different from the accepted opinion (“doxos”).

The key thing to note here is that these businesses are choosing to have their computers placed in a place that hosts them together with the computers for other businesses.  These computers share the same location (or are located “next to” each other in such a center).  “Collocation” indicates such “locating together,” as can be seen by the parts of the word.  “Col-” comes from the prefix “Co-” or “Con-”, meaning “together with” and is found in words like “colloquy” and “cohabitate.”  The second half, “location,” is obvious in its meaning.

Although any of these options apply to certain kinds of mental failure, the best option is the general one—“senility.”  This likewise can apply to the general loss of strength in way that the others cannot.  The word “senility” first indicates the general signs of aging and weakening from that.  It does especially apply to loss of mental ability, though not solely that.  The word is derived from the Latin word for old man, “senex.”  It is related not only to the English “senile” but also “senior,” as well as “senator.”  The last title was chosen because such lawmakers are intended to be the “wise old men” of the country.

Since the foreign government wishes the United States to send the man to their country for trial, such an act would best be called an “extradition.”   To “extradite” someone is to send them to be tried in a country in which they committed a crime.  The word is a compound of the prefix “ex-,” meaning “out of” (found in many English words like “exit,” “exfoliate,” and “expropriate”) and the latter portion “-tradition.”  This latter half comes from the Latin for “to hand over” (itself a compound), as can be seen in the very form found here “tradition,” a word literally meaning “something handed over (e.g. from generation to generation).” 

The sense of this sentence is that drinking too little water is connected with the occurrence of migraine headaches, although we cannot say if there is a direct causality.  When two things at least occur at the same time (even if you cannot claim causality) they are said to be “concomitant,” a word meaning “occurring at the same time as something else.”  Think of it as meaning “sharing the same company (-comitant) with (con-) something else.”