So, here's a quick summary of my question: Is it possible to use your GitHub Pages domain for anything that isn't a website? Say I have a server that hosts an API, and I need a domain for that. Is it possible to use my GitHub Pages domain for that server?
No, you cannot do that.
GitHub pages is a static site generator. It generates the pages from your GitHub repo and serves them. You have no control over how the pages are served.