Nice job getting organised in advance and taking the initiative with questions you'd like answers for. =)
There's a bit of crossover with the "Everyday UX" ebook where Matt and I were looking for similar things - to get an idea of what the working day looked like and how preferences and skills influenced how a UXer actually did their work, not just their hard-skills but also the personalities and approaches behind the professional and how it went together as a package. The kinds of questions Matt and I asked were mostly aimed at getting to know them personally within that professional context.
I guess workplace culture is not dissimilar to a company-wide version of that. Here are some quick translations of a couple of questions, feel free to use or modify as appropriate:
- How/when did the firm first get involved in user-centred design or UX?
- How has the approach to UCD/UX changed since it started?
- What kinds of projects are ideal for the firm, or most enjoyed by the staff here?
- Which UX techniques are most commonly used here? Which techniques aren't used so often? Why?
- What does a project team look like? Who works here and what are their skills? What skills are they short of?
- How does the firm define a good user experience designer? Specialist, generalist, research-focussed, project manager, code-savvy...
- What are the options for professional development and ongoing learning?
- What are some of the biggest challenges faced by the firm in project work?
- What does a typical day look like? Tasks, responsibilities, types of meetings, tools, documents, lunch, etc
I'm not sure how well you know them, or how much they'll be willing to share. Hopefully that's a helpful starting point?