How much authority will I have in running the department (the group, the facility, etc.?)
You already may have picked up on a lot of the authority you may have in the interviewing process. But, now you need to get a real, detailed understanding of your authority. Having P & L responsibility usually carries a lot more responsibility than not having it. Don't assume because you think you heard it in the interviewing process. You were focused on selling yourself then, now you are focused on qualifying the job, your expectations for the job, and your ability to do it.
I would like to speak to peers about the position that I am interviewing for. Can we arrange that?
Unless the organization that you are interviewing with is replacing somebody confidentially, any firm worth its salt is going to allow you to talk to other managers that are in the same position you are applying for. Make sure you do this if you haven't done so already. (And reread the section above about interviewing in groups and with potential peers.)
You'll get more insights about the company and the job from your potential peers, especially in a management position, than you will from anyone else in the company, even the people you were interviewing with. You'll probably be able to pick up undertones of political issues, personality problems, and organizational challenges by talking to these people. But it is imperative that you do!
If the organization does not want you to talk to other managers, it should raise a big red flag to you. But even if filling the position is confidential and they're replacing someone without their knowledge, a hiring authority should be able to provide you other managers who may talk to you on a confidential basis. Don't underestimate what you'll learn by doing this.
This question is also one of timing. If you asked this question too early in the interviewing process, you will certainly not be accommodated. But once the organization intends to make you an offer, and it has invested in you as a candidate, you have more leverage and are more likely to get audiences you request.
Are there any difficult personalities on the staff that I will be supervising?
Pay attention to the hiring authority's initial response to your question. If she hesitates and has to answer the question carefully or pensively, you know that you might have real challenges here. Let the hiring authority describe to you all of the people on the staff that she knows about.
People get really uncomfortable when they're getting a new supervisor. So, you need to be prepared for any difficulties, especially difficult personalities, that you may inherit.
Are there any members of the staff or staffs that should be let go?
Oftentimes, if a new manager is being brought on board, people who ought to be fired or moved around are left in their places until the new manager is hired. Upper management will insist that there is enough turmoil going on with hiring a new manager and simply leave the difficult task to the new hire.
Be ready for any kind of answer to this question. If there are really big problems—for instance, a whole department needs to be replaced—you will probably have heard about it in the interviewing process before now. But now it is time to press on this.
If you have to replace the whole staff you inherit, your success will certainly take longer. Normally, you won't have to replace a whole staff. But, unless you are very lucky, you will probably have to replace some of the staff. You should assume that these people should have been let go a long time ago, and their dismissal has been postponed because it is hassle enough to find a new manager. So, listen well.