The questions you ask to understand the culture of a company and design team are contextual to what you yourself value. Jot those down. Your questions should help answer if an organization exhibits the traits that you value.
1. Figure out the values that are most important to you for your next opportunity.
Is it collaboration? Is it a team that values people uplifting each other? Is it an inclusive work environment with a diverse workforce? Is it a company that values employee growth? Is it where colleagues regularly get together outside of work? Is it a company that's mission-driven? Jot down what you value. Ask questions that help answer if the organization exhibits the traits you value.
2. Ask about process and how designers interface with cross functional peers.
While it's important to know about the design team, you're likely going to work closest with your cross-functional peers (my advice here is primarily from the lens of working at mature organizations. It's perhaps different for newer teams). What you likely want to find out here is what the designer to product manager, designer to engineer, designer to insert-other-cross-functional-peer working relationship is like. You also may want to ask questions that help answer if designers are involved in product strategy and if the Design to PM relationship is a true partnership where design is coming in at Quarterly Planning, or if it's more of a waterfall process.
3. Find out if a company has a set of values.
A lot of companies have these. Both Lyft and Facebook are great examples of organizations that have a solid set of values that are shared and explicitly listed. Once you've found out what an organization's values are, you can ask questions that help answer how it permeates throughout the organization, how/if it helps to drive decision-making and its impact in practice, and how the values shape the organization. What you want to find out is if the values are contrived and there as lip-service (or because all the cool kids are doing it), or if the values are there to shape and guide the organization. The idea of "culture-fit" is often perceived to be a euphemism for not hiring people that aren't like the hiring committee (and often negatively impact marginalized groups), while "values-fit" recognizes the importance of having different voices while maintaining an inclusive and ever-evolving culture.
4. Figure out how inclusive the product development team is.
Sometimes you can get a sense of this from walking through the office (typically, you'll get a tour of the office before your interview starts or at some point during). While I believe diversity is multi-faceted and goes beyond gender and ethnicity, I also believe that an organization where you can visibly see diversity values inclusivity more than organizations where it's not evident. In my experience and the experience of friends, when it's not visible, when there aren't many women in leadership positions, when there aren't many women in product development (design, eng, pm), the organization is more resistant to a diversity of thought, is less collaborative, and bad behavior may be rewarded, forgiven, or overlooked.
5. Look at how inclusive your interview panel is when you're at an onsite.
Similar points as the above Action Item. These are the folks that will be evaluating you. How diverse is it? (Reiterating my above point: diversity is multi-faceted). Keep in mind if the team or org is newer/smaller, they may value inclusivity but are perhaps not yet at a point where interview panels are visibly diverse. If it's a more mature organization, then it may be acceptable to be more critical.
6. Ask questions that help you figure out how designers interface with each other.
Are designers silo'd from each other? Or do designers regularly interface with each other on some regular cadence (weekly design crits, bi-weekly team crits, etc). Asking questions around team structure will help you understand how designers are allocated to teams. Asking questions around collaboration will help you understand how designers from different teams work with each other (often times, decisions made in one team can impact another team, so collaboratively working together is important to ensure positive impact).