The English translation is grammatically incorrect and it should be either "Why didn't you/why don't you give me your book?" It could also be "Why aren't you giving me your book?"
How is this used, exactly?
Is it more like a suggestion or an actual question?
"Say, why don't you do this?" versus "Why are you not doing this?"
It can be used as both, but I think the Danish one might suggest more the latter, but to be honest it could easily be both depending on how the sentence is stressed. :)
It depends on the context - it can be used like: "Why do you not want to..." or "What is the reason that..."
Isn't ikke supposed to follow the verb here? why is it placed so randomly in the sentence? Please help me understand the placement of ikke in various sentences
I think the not is placed there because the object pronoun is technically part of the verb because Danish love phrasal verbs. The closest example in English I know is the dative imperative, "Forget me not, Lord."
"why aren't you giving me your book"?
why you dont give me your book? why is it marked wrong?
Because it is 'why don't you ...'
The translation is wrong : Why don't you give your book