"Whose book is it?"
Translation:Vems bok är det?
det is not an article of the book here, it is just a placeholder pronoun. We often compare it to it in it is raining. Just like we say Det är min bok 'It is my book', not den, we would only use det here.
For a much longer explanation, see this post: https://www.duolingo.com/comment/9708920
As I guess you figured it out by yourself within the period of time that elapsed until today, I can only tell that it sounds very German to me. Hence, I would state that the “Vet du” is a disposable particle placed in front of the actual sentence, which functions in accordance to the regular SVO structure. The aforementioned particle is not necessary in order to construct a complete sentence, hence the different positions of “är” in both sentences.
The "inversions" presented in the notes have been very helpful to me in understanding question structure (I take a statement and then "invert" or switch the noun and verb to transform it into a question), but I can't seem to transform this question into a statement to work backwards. Can someone help?
Actually talking about inversion may not be that helpful. The rule for main clauses is that the verb is in second place. While in most cases the subject is first, it doesn't have to be and there's no rule about it. But the rule about having the verb second is very firm.
For questions, the rule is that the verb goes before the subject (and you can only have question words etc first).
The reason this one is hard to rewrite is probably 'vems'. It's a question word, so we wouldn't use it in a statement.
With possessive pronouns, it would be like this:
Det är din bok. 'It is your book.'
Är det din bok? 'Is it your book?'