What questions should we be asking a care facility before choosing one?

I suggest that you ask the Social Worker from Hospice what facility they have a good working relationship with.
they may not be able to completely recommend one but they might be able to steer you clear of less desirable one.
Oh, another advantage asking Hospice...they may be able to get him admitted to a facility that accepts Medicaid faster. (not to be harsh but if a Hospice is looking for a bed the facility probably realizes that this is not going to be a long term resident.)

Here is a link to a list some of us made.

The question we were answering was about a MC facility, but many of the concerns were the same.

https://www.agingcare.com/questions/questions-to-ask-memory-care-facilities-before-placing-a-lo-470196.htm

Medicaid is what would pay for his long term care in a facility that accepts Medicaid recipients. Your FIL will need to qualify both medically and financially. Someone with access to his informatiion would need to fill out the application for him. It takes about 3 months to hear back from them. Each state has their own rules.

If he is a horrible man, you can report him to social services and allow the county to acquire guardianship of him, then you don't need to deal with any of this yourself.