What are some questions to ask a hospice nurse?

You should ask the charge nurse the questions to ask.

As I have said here before I would not have been able to care for my Husband the way I did without Hospice.
Ask the Nurse for any information she has on the End Of Life process. Most will have a pamphlet that they give out. It contains a LOT of information and to know what is expected sort of makes it less frightening.
The Hospice that we were with was wonderful and they have a goal that no one dies alone. So when the end is close if you can not be there I am sure they will have someone sit with him during the times you can not be there.
there are also Music Therapists that can come in and play music that will calm, soothe and relax him.
As far as what to ask...
There are no questions that they have not heard before.
Typical is "How long" and that is difficult to answer but they can be pretty good at guessing depending on skin color "mottling", breathing, stopping eating and drinking is one of the first things that will happen. When they stop eating and drinking pain medication is usually done under the tongue with morphine as the 1st choice. Don't let the use of morphine scare you. It is safe. The use of morphine will NOT hasten death. Very low doses are used to begin with as the body gets used to it or as pain increases the dose will increase but that is just like any medication.
Also with morphine the body will relax. Breathing will be easier. (after I gave my Husband the first dose of morphine I was able to easily move his arm that had pretty much been locked in place for years. The movement was easy and relaxed. He also slept peacefully.)
Hospice is wonderful and the sad thing is your Dad may have qualified for Hospice a year or more ago and he would have had more individualized care from the Nurse and a CNA weekly. Many people think that Hospice is just the last 6 months or less but as long a person shows a continued, documented decline they can remain on Hospice. (My Husband was on for 3 years)
Tell your Dad that you love him..you will be alright..and that it is alright for him to go. He needs to know you will be fine, you will be sad but it is selfish to want him to remain as he is, he is not the vibrant Husband and Dad that he was.

I'd ask why he was put on hospice. Had something changed that was an indicator that he was entering the dying stage?

If they haven't offered it yet, you might think about asking for music therapy or massage therapy; massage can be as simple as a gentle foot rub and is very relaxing for patients. Also, chaplain visits (Chaplains don't necessarily talk religion unless the patient wants it. They can offer kindness and a warm presence). Also, social workers in hospice are great at helping with logistics and any financial concerns. Bath aids two times per week are available and make your job easier. Don't be afraid to ask for a change in meds if what you have isn't helping. Hospice nurses can offer just about any med needed to make sure the patient is comfortable. Also, almost any supply you can think of would be available through hospice, so don't be afraid to ask.  Last, having a volunteer come in just to visit or sit and read to a patient is another option.

Thank you both. I appreciate the help and advice.

We also had a wonderful hospice nurse that was like Blannie’s experience. Made an awful time a little easier to bear.

When my mom was on hospice, the nurse was very good at helping me understand where mom was in the process and what signs to look for as mom was coming to the end of her life. She talked about her temperature, her breathing, her coloring, etc. So I'd ask her what to expect as your dad transitions into the different stages of the dying process. I found it comforting to understand what was happening to mom and to know that what she was experiencing was normal for a person coming to the end of their life.

She also helped me understand the medications they prescribed and why they were used. For example, they prescribed morphine, which I thought was only used for pain and my mom wasn't experiencing pain. My mom had lung issues, so the morphine spared her the feeling that she couldn't breathe well because of her pulmonary fibrosis. They also gave her Ativan for her restlessness. She explained those meds and what they did for mom.

The nurse was able to predict that my mom would pass on a particular day and she did (near 11 PM that evening). I hope you get a nurse who was as caring as the one my mom and I had. Hugs...