What specific skills from the person you hire would make your life easier?
This question focuses the conversation squarely on the proposition that the employer has a problem. As the potential new hire, you want the employer to tell you that you can make his or her life easier because your skills are just the ticket.
How does upper management perceive this part of the organization?
The response to this question will give the job seeker a feel for how valuable the department is to upper management because if and when the organization goes through a financial crisis, you want to know that your department will not be the first department cut.
What challenges might I encounter if I take on this position?
Listen carefully. The hiring manager is telling you where you are expected to fail. Is this a challenge you can take on and at which you can reasonably hope to succeed? If Superman couldn't hack it, watch out! You're being set up for failure.
What are your major concerns that need to be immediately addressed in this job?
Note the emphasis on the word “your.” This is less about the organization's agenda than the hiring manager's concerns. They may or may not be different. It won't serve you well to meet the organization's goals but not your manager's.
How does the reporting structure work here? What are the preferred means of communication?
This set of questions goes to the heart of the corporate culture. Are reporting structures formal or informal? You will not be happy if you prefer an informal, open-door company environment and this company prefers a more rigid structure.
What areas of the job would you like to see improvement in with regard to the person who was most recently performing these duties?
This should give you a clue about why the incumbent failed. Yes, it's true that people can learn only from mistakes, but that doesn't mean it has to be their own mistakes. The downside is that if the incumbent left on bad terms, you risk associating yourself with some negative vibes.