Translation:This book is not very thick.
Don't forget 熱い for hot (thing) in addition to 暑い for hot (weather), both pronounced as あつい
"Hot" was among the options, and I fell for it. "Why would you expect a book to be hot?" xD
I disagree. To me, "あまり" simply has the connotation of "much" (both in magnitude and frequency - other words to illustrate its meaning include "very", "a lot", and "often"). The word "so", on the other hand, has a connotation of comparison, implying that some other book is understood by the speaker and listener to be thicker than it.
I disagree. "So" is used in both of these ways. The comparison requires context, the noncomparison is colloquial.
To continue off of honeyrbw, ない and ありません are the same word. The former is the informal form of ある while the latter is the formal form. As for why です is used, my guess is that it adds some formality (or politeness) to the adjective, but using ありません is the highest form.
"Atsui" is an adjective meaning "thick", so using "thickness" (a noun) has a similar meaning but is not quite the same.