Do I have to purchase a commercial license for every user in our team?

If you have chosen the commercial licensing, each Qt user must have their own assigned Qt commercial license. UI Designers, Technical Artists, Software Developers, Test Automation or Quality Assurance Engineers might have different kind of Qt licenses, but every single person needs a valid commercial license.

Also, software developers in one software project must have one type of commercial developer licenses. For example, developers working on same project cannot use both Qt for Application Development Professional and Qt for Application Development Enterprise licenses, they need to choose one license type for all team members. 

It is not permitted to mix Qt commercial licensees with open source licenses. The Qt Company has the right to audit the use of commercial Qt products.