Certain nouns present the same version for both genders, (e.g. ο/η συγγραφέας (the writer), ο/η οδηγός (the driver).
For some others, the same version can be used for both genders (ο/η βουλευτής - a member of the parliament, a congressman), but there is also another version exclusively for the female gender (η βουλευτίνα). All of them correct.
Finally, there is the ones that always alternate according to the gender (e.g. ο δάσκαλος/η δασκάλα - the teacher)
I am afraid you will have to dig hard to the archaic roots to get a rule for this. In most cases you will just have to learn these words case by case. But do not get frustrated, lots of greeks find this hard too. ; )
I agree. Your suggestion “The author writes a book” would be a better translation for English speakers. However, the translation above is not too unusual. Consider for example, “The bakers baked the most delicious treats” and “Her students studied harder than they ever had before.” There is a literary tool known as alliteration that aims to produce repetitive sounds in poetry and other works. According to this Wikipedia entry, alliteration is a type of parechesis (παρήχησις): the repetition of the same sound in several words in close succession.