To answer your questions in reverse order...
You and your daughter lived in Arizona. Presumably, when she finishes school, she will come home to Arizona. In this case, she lives in Arizona, and the stay at college in Missouri is considered temporary.
She would not need to file a Missouri return unless she took a part-time job (or full-time) in Missouri, which you say she hasn't done.
Now to the first question.
It is not normal to file a return for a dependent who has no income. Not normal does not mean illegal, but just not common.
It is much more common for the parent who is paying for so much of the school burden (tuition, supplies, books, etc.) to also file for the educational credits. These do the parent with the higher tax bracket much more good than the student with no tax.
Honestly, I would think that this would benefit you more than $500, but, of course, I can't know that without seeing your return. But the education credits can be much larger than that. This is in part before the American Opportunity Credit is only partially refundable and the Lifetime Learning Credit is not refundable at all. Non-refundable credits do your daughter no good at all, but may well do you a lot of good.
I assume that the $500 is the Credit for Other Dependents, but please go look at you taking the education credits.
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