Coukd you just write "kardeşim yazar olmak istiyor"?
Not really...that would just imply that one of your siblings does. It doesn't specify that it is your brother.
How do we know when to say "ABIM" and when to say "ERKEK KARDESIM" ?
Erkek kardeşim yazar olmak istiyor is also correct, because we never say "kardeşim BİR yazar olmak istiyor"
This is already accepted and is the best answer :)
What does olmak mean in this sentence
It means "Become"