Do we really need the在 (zai)?
在 here implies the present continous tenses, so it should be included.
Is there anything wrong with saying "Ta weishenme zai ku?", or does the question word have to be first in this sentence for some reason?
她为什么在哭？is good too
她为什么哭 is okay or no?
It would be "Why does she cry?" or "Why did she cry?", so it depends on the context.
What's wrong with 她为什么哭?
在 implies the present continous tense, so your answer technically translate to " Why does she cry?". Yes, grammar sucks, but teachers are usually extra-harsh when you're learning a new language.
They accepted such a phrase ...right now
Shows how little I've grasped. I expected a 了. She wasn't crying before but that's changed and now she is.