1. Can I see references from the previous clients?
Most of web development companies have testimonials right on their sites. If not, you can ask for customers' contact information but be ready for refusing as there is a possibility that company doesn't have any arrangements with customers about sharing their personal data. In this case you can look for telephones and emails by yourself.
2. What are your payment terms?
Usually you have to be ready to pay 50% in advance, and it is quite weird if company agrees to start working without prepay. Professional web design studio just can't afford to spend time and resources without being sure that the client will continue working with the company. They may ask you to pay the rest of the sum just before the end of the project. Payment conditions depend on project and are usually negotiable and flexible, and you will have several different options to fulfill payments.
3. Can we make changes during the work and later?
You have to know from the very beginning when you can ask for changes, and what changes it is possible to do at particular stages of the work. Discuss the financial aspect. Some modifications may be easy to perform while others might need a lot of time and effort so company can ask for extra payments. Usually web developers overestimate time needed for work as this gives them a little time to do more tests or correct mistakes if something goes wrong. Some changes you will be able to do by yourself. For example, site management systems usually allow to change texts, add images and pages.
4. What are options for SEO?
This is a very important question! Site must be ready to meet SEO requirements to get high positions at search engines. Each page should have unique title, description and key words. Quality content is critical as it is one of the most important ranking factors. In most situations web studios offer clients with copywriting cervices so you don't have to think about writing texts. Don't forget to ask if the CMS (content management system) allows to set SEO titles, meta descriptions and key words while creating new pages.
5. How will we measure results?
If you are ordering site design, the answer is quite simple. Good result will mean that site is created on time, everything works correctly and all your requirements were considered. When you ask about site promotion, there should be more details. The company have to install analytics tool at the site and provide you with reports about site's positions at search engines, conversion rates, number of visitors and time they spent on site, most viewed pages etc. Make sure you hear all these phrases when ask about web promotion.
Hope these questions will help you to make your work with web studio effective and pleasant. Good luck!