Some great responses here. I would also be direct in asking about their core values. What values rule their company? What values do they use to evaluate decisions.

I would also test them in a smaller capacity to ensure that they:
1. Are exceptional communicators
2. Adhere to deadlines
3. Exceed your expectations of quality.

You might do this by hiring them for a smaller project. We refer to this as "doing a little business so we can do more business."

One thing you will want to understand is their process. How many phone call interviews with you will they engage in before starting to actually design? How deep do they go? If they are going to talk to you once or twice and then start designing, that's a very different process from a team who is going to immerse themselves in your culture and conduct 10 hours of interviews before they start their work.

Ultimately, you want to feel a strong sense of alignment with them emotionally. A branding firm has to distill the essence of your organization and then package that visually. So they have to be able to dive in and get to the heart of your organization.