Why can't it be, "you are writing a book"? How can you tell it's meant to be a question?
Because there is a semicolon (;) at the end of the sentence, which is what the Greek question mark looks like. ^.^
The correct (literal) translation is 'do you write books?' In reality, a native english speaker would probably ask 'are you a writer?' Or do you write (adjective) books?'
Weeell, if it was the first book that I am writing I would say I write books neither that I am a writer xD
Duolingo only allows one answer to be accepted in listening exercises - no homophones. :(
Also, I don't think it makes sense. Maybe "Γράφεται το βιβλίο;" as in 'it's getting written by itself', but it's still strange without further context for the passive voice. "Πώς γράφεται το βιβλίο;" for example would be valid.
Because there is no need for the definite article here. This sentence's meaning is general (since the article is omitted), so only the indefinite article would work. ^.^