I have created a CTP called document with Built-in Taxonomies Category (WP-core).
Still, I can’t choose the primary category when the document CTP has more than 2.
Also, the permalink only display: http://www.mydomain.com/document/my_document_title/
The category slug is missing.
All of this is generally default WordPress behavior, and I don’t know for sure if there’s any plugins to help with the primary category part. One of the issues that you’d run into there is duplicate content possible which Google doesn’t like. That said, there could be solutions available.
Regarding the permalinks, another default behavior for WP as posts are considered special and get extra treatment out of the box, including permalink setup. I know there are plugins available to customize permalinks, but it’s not something we have built into CPTUI.