inspired by a love of birds, Nancy Yi Fan (has begun to write) her novel ‘Snowbird’ when she was in fifth grade.
a)same
d)began writing
why doesn’t has begun work in this case? is it because the action was presumably ended in past? can i get explanations for difference between has and simple past?

If she’s not still writing the book, then you don’t use “has begun.” Rule of thumb: don’t use perfect tenses (like “has begun”) unless there’s something in the sentence that really calls for it. Make it your default reaction to reject choices with perfect tenses.