I suppose its always good to ask about structure. How meetings are held, how the department is managed, what are the dynamics, etc. I find it to be something that evolves according to certain ideologies, and being on the diapason with them on their communication channels can't be bad.

I would also suggest reading any review related to your prospective company that are posted on GlassDoor or JobVent. Naturally, most of the reviews are negative, but you can sometimes identify common complaints about the company and this can provide you with some insight in the real company culture and give you some questions to (tactfully) ask. They also have a section of interview questions.