To generalize, what's the user interface design process for new Stack Exchange sites and how much visual and functional flexibility is allowed?
The unique designs of Stack Exchange sites are created by SE employees only. The design workflow does not allow moderators to tweak graphics or make wholesale changes, as they might be able to do with say, a WordPress site.
The Stack Exchange employees who do this have the title of Product Designer, and occasionally may be noticed discussing their proposals using the standard Q&A format. While the Product Designers create the design and have final say, they may also work collaboratively with the community on the concept, design language, and fine tuning of the final product.
Some design elements can be always be changed, some can be requested, and some explicitly cannot be changed. As @JoshP helpfully noted, at least one meta post provides some of the details, which are spread across few different answers (don't stop reading after the first answer).
Sites with special requirements may actually request new capabilities not possible with other sites. This is a tough request to grant and it doesn't happen often, because it can involve a significant amount of work including custom coding. However the Stack Exchange team does listen, and weighs the specific context as well as whether such functionality might benefit other sites now or in the future.
The best explicit references I've found are the link mentioned above, and a targeted google search that shows examples of many different design proposals/discussions.
Design parameters for new SE sites:
Examples of design proposals and the resulting community negotiation: