In particular, I'm looking for information such as the filename of the template being rendered. However, in general I'd like to know if there are any automatic global variables available to the template. I couldn't find any information in the documentation.
You can see all available vars in current context - https://stackoverflow.com/a/38881808/6121703
Template can be render from web-url (see WebLoader) or string. So there is no filename, but source. In any case I don't find simple way to get this info.