Can my employer ask for verification of my vacation such as reservation and cancellation policies? What if I am going to see my father in prison or my sister at the betty ford clinic? Am I required to tell them that?
I realize they can deny my vacation request but can they ask for this kind of personal information?

Yes they can, unless they have written policy saying otherwise. There is very little an employer can't ask about that is about current, even medical conditions. Vacations are a benefit, a gift, and some employers even want you to be reachable or recallable or they limit when you can go. Health situations affect the company and the other employees (TB or Hepatitis, e.g.)
This is not to say you 'can't' lie or omit or be vague, but you will need to take the consequences if found out. Your best tactic is to trust the one person at the top who needs to know and tell them that you trust that the information remains at that level.

Sure they can ask, but they can't require it unless you are an on call person. If you are in a position where they may need to contact you and you are telling them, that you will be out of touch, then they may need verification.

The fact that they are asking about cancellation policies, indicates to me that they may want to know if they are calling someone in, that it will be the person least disrupted.