if 'var' asks for the direction of something ("vet du var vargen går?"), why is it not 'vart' since the woman knows the books location.
It’s the other way around. ”Var” is used for location, and ”vart” is used for movement. The book is lying still, so it’s ”var”. If you say ”var vargen går”, the wolf is walking around in a fixed location, it’s not moving to that location, so you’d still use ”var”.
That being said, many Swedish speaker don’t make the distinction and use either ”vart” or ”var” for both, but it’s good to start by learning the difference.
I thought that in those cases "där" and "dit" should be used, i.e., "Kvinnan vet där boken är.
I see the verb "vet" is unusual since it doesn't end with "er or "ar". are any more verbs like this?
Vilja is at least one example. Vara ends in -r in the present, but is also irregular. I'm sure a native speaker could give a more comprehensive list, but they're out there
It should be but Swedish pronunciation tends to be very relaxed in normal speech.