Does γράφουμε always mean compose? Or does it also mean to write?
Γράφω primarily means to write. Compose is συνθέτω, but in Greek most of the time you'll hear γράφω in its place for music, e.g. "he wrote this song/track/symphony".
Why is it not, "We are writing a book"?
That should be fine.
Why the a in this sentence but no a in δεν γράφω βιβλίο or in this one: γράφετε βιβλίο;