How to use EXT:felogin?

The usage of felogin with headless is not different from the common setup following the felogin documentation.

To test the login without frontend, we can use browser extensions like RESTer (Firefox) or Insomnia REST Client (Chrome) to simulate form submission. If the response contains a set-cookie header, the login was successful.

Does EXT:headless work with other extensions?

Yes, basically all extension output can be rendered into the JSON response. Read the integration of external plugins section of this documentation, or take a look into the code of headless_news which can be seen as example how this works.