Is there a REST(-ish) API that I can use in javascript in a theme that I'm trying build? With that, I could do some more fancy stuff.

I couldn't find anything like that on the user guides, nor anything useful on the forums, so I'm not hopeful. But if it does exist, please let me know where to find documentation for it, and how to access its endpoint.

No, we don't have such an API and none is planned in the near future from our end. For sophisicated functionality outside the "normal theme ways" there is the base object model.

Maybe take a look at the externalFeed plugin you maybe can do some similar things with.