Translation:When is his birthday?
When's his birthday is the exact same thing as when is his birthday......
Why is “她” rejected? Isn't it equally valid in the absence of some other indicator?
Absence of some other indicator ? "HIS birthday". Now if you were only given the audio I could understand your question but I never came across an exercise on this course where you get an audio sentence and have to write it in Chinese .
Why would you use 时候？日期 makes more sense. 时候 is like time. Birthdays all start at the same time: midnight, so what's important is the date. Now if the sentence was about a party, it would seem normal but this just sounds weird to me.