The Coda Community is not build on Coda but on Discourse so, how the Categories list looks has nothing to do with Coda …
(Which doesn’t mean that something similar can’t be build in a doc though, using a card display, some icons (simple PNG images stored in a Image field), some conditional formatting, etc… and while it won’t look exactly like the Categories list in your screenshot, it could probably come somewhat close)
As for your first question, if you click on your picture in the top right corner, and then, on the “people” icon …