I sort of expect that candidates coming for campus visits will have looked over the website and found some of the stuff we brag about here. Asking about stuff we brag about is enlightening. For example, say we brag about undergraduate research. If you ask faculty folks about undergraduate research opportunities, you'll learn a lot. If we give you a blank look or laugh in your face, you'll get some ideas about the disconnection between the administrative bragging on the website and our experience. But if we talk about trying this or that, or a colleague who's had some experience, or a student who's done something really special, then you'll know it's real, and you'll learn something about how we do it.
If there's a special department or group project, asking about that's likely to be informative. For example, if you were interviewing with a biology department that has (and advertises) a special relationship with a local prairie restoration project, or a chemistry department that does some cool air quality thing, asking about that will tell you lots, and you'll also likely impress your hosts.
The harder things to get at are things like benefits, quality of life, equity issues, and so on. How do you ask about maternity issues if you're worried that the department won't hire someone they think is likely to get pregnant? I think that's a really hard one. I don't really have an answer.
My hope is that someone along the way will talk about their kids, and that will give you an opening to ask about local daycare, maternity leave, and so on.
If you're openly gay and the department is even minimally aware, then it's probably not at all a problem to ask about partner benefits. (Unless, of course, you're interviewing at a religiously affiliated school with an unfriendly affiliation. Not all religious organizations are unfriendly to gay and lesbian folks, of course, and it's good to get a sense of that earlier rather than later.)
I would tend to ask hard questions in terms of "what was your experience with X" rather than asking for a policy statement or something. So, if you're out with tenure track faculty, asking someone how their experience of the review process has been seems reasonable. If they burst into tears, then it hasn't been good. Let's hope that doesn't happen.
You might also ask if they feel well-informed about what's required for tenure. If they all feel well informed, but each tells you something totally different, that's useful information. (I know of a department where, when asked, each of the five junior folks had a rather different idea of what was required beyond "good teaching," especially in terms of research: from a single article, to X or Y number of articles, to
You might ask what sorts of resources the department has for travel for research or conferences, and how it decides how to spend it. (My department, for example, quite openly supports tenure track faculty with money first, and then tenured folks get less. We all pretty much think this makes good sense.) If you're in the sciences, you'll want to know about lab start up funds and such. (No one cares about us poor humanists and our needs, waaahhhh.) You can ask about the library, and what sorts of support it offers to faculty for teaching and research.
If you have a partner or kids, I tend to think it's a good time to ask about local schools, opportunities for work in your partner's field, and so on.
I've thought a lot about the difficulties of this community for people of color over the years, but I wouldn't begin to know how to help a candidate who's a person of color to negotiate those questions. (For one very obvious thing, pretty much every person of color who's likely to be on a candidate visit has a whole world of experience more than I ever will.) I think at my campus, the administration has looked at our record in hiring and retaining faculty, recruiting students, and graduating students, and seen that we're unimpressive on all counts. At times, I've felt the administration was making real efforts to change things. But then administrators come and go, and the next wants to focus on something else, and lower down the chain, funding for retention programs and so on dries up, and we're back to square one. I don't know how visible that process is for a candidate.