1. Pricing: Ask every web design firm about what goes into their Pricing Structures?
Ask them for billable & Non billable hours, Meeting versions.
2. Portfolio: Everybody should ask about Examples for past web design work done. If you do not see it, Probably it doesn't even exits.
3. Coding: Ask them for SEO friendly & Responsive Web Design Code. A well coded website with specific SEO elements is what you need from your web design firm.
4. Reputation: Check their reputation as "Do they have necessary credentials?"
After that check their LinkdIn, Facebook and Twitter business page.
5. Copyrights: This is the major one, Who Owns the Work?
Some firms will try to license a site to a customer instead of transferring ownership. Both models can work, but be sure you know what you are getting for your money.
6. People: This question about 'How many people will work on your site?' will always comes to your mind, So clear it with your web design firm. A large scale team has advantages over a small scale team and vice versa. Larger teams may have faster turn-around times, but more issues with communication and getting the design done correctly. So ask them about this serious issue.
7. Sustainability: Is the firm busy enough to Sustain the development?
This may sound like an odd question, but a company with least work may be at risk of shutting down in opposite companies that take so many projects make a project delays.