I've just taken a course on History & Theory! I swim in cultural and intellectual history so I found: Geertz, Foucault, Said, Darnton, White, and La Capra to be the most useful for me although we had to read many others in my course.
I would call "crucial" anything that makes clear that empiricism isn't the only avenue to producing quality scholarship.
She very likely could serve on your thesis committee. In my correspondence w/ OSU it was suggested that I also reach out to a prof in the English department because of my interests so it seems as though it's encouraged!
If I'm not mistaken, Alexander is a law professor there but is affiliated with the Kirwan institute for race & ethnicity, which isn't exactly a department I don't believe, so you'd have to select an alternate prof as an advisor. I'd recommend Prof. Hassan Jeffries from OSU in the history dept.