I wrote "the author types their new book" as there wasn't anything in the sentence to indicate gender. why didn't that count?
As well as that, the translation at the bottom of the screen says "her book" but the translation here at the top of the comment screen says "his book".
1) I have added your suggested translation now 2) The reason it showed "her" is because when you make a mistake, Duolingo shows the closest correct translation to what you typed (it considers "her" closer to "their" than "his") "His" is shown above because that is what is marked as the top translation.
It wouldn't. The grammar depends on the gender of the word, not the gender of the person.