I work for a property management company and these are the questions I most commonly hear from owners who are "interviewing" us.
1. How long have you been a property manager?
2. What types of properties do you manage?
3. How long on average does it take you to fill a vacancy?
4. How does eviction process work in your company?
5. What are your management fees?
6. How long do I have to have a contract using your service for?
7. What is your repair process?
8. When will I receive my rents?
9. Can you use vendors I use?
10. How many units do you manage?
Most of these are pretty basic that you can find online as well and they cover most of what you need to know. However, I think you should always ask about the insurance that the company carries, units managed to property managers in office ratio, how they determine rent pricing on vacancies, how often they communicate with you, is there any hidden fees (late payment charges, court appearances, leasing), if you can cancel contract if unsatisfied, provide monthly statements, any references. There are a lot of things you can ask but these are just some off the top of my head, so you can see which you feel is more important.
Just remember, you are entrusting your property to someone else. So when deciding a PM company to use, you are the one to decide what you are most comfortable with and what you are not.