Translation:That textbook is not thin at all.
You're actually asking about the difference between 薄い「うすい」 and 細い「ほそい」. I believe うすい is "thin" in terms of being more flat (like a paper or book), while ほそい is "thin" in terms of being more long and narrow (like a road or a rod).
The く in うすくありません is part of the negation portion (くありません). Which is a more formal version of くない.
It's funny, because this course (JA, from EN) usually marks answers as wrong if you type "the" for その, whereas the reverse course (EN, from JA) often marks answers as wrong if you type "that".
I often forget which course I'm currently on and end up typing the wrong one. Gets rather confusing that the two courses each have their own default preference for how その should be translated. I'm always relieved whenever the sentence has あの instead, since its default translation is the same in both courses–"that". xD
Yes; in fact that's pretty much the same thing as the informal negative of ある is ない. About negative adjectives in general, you have to understand that there are two kinds of adjectives: i-adjectives like うすい and na-adjectives like 大切な. To make i-adjectives negative you chop off the い and replace it by くない／くありません (you can think of that as うすく being an adverb acting on ない／ありません if that's easier for you, "x doesn't exist thinly"). To make na-adjectives negative you chop off the な and add じゃない／じゃありません.