"That woman does not know that book."
Translation:그 여자가 그 책을 몰라요.
Its's not really negating per se. 몰라다 is the verb: to not know. I suppose you could say 안 몰라요 to say "i don't not know" = I do know.
No, it is 모르다. Since the stem ends in 르, it is dropped and turns to 몰ㄹ. Since the last vowel is ㅗ, ㅏ is used and the final form is 몰라요.