Is there a way to use jsTestDriver in a headless state where it loads the browsers but does so headlessly (you can't see the browser) or it loads them minimized?
There is no way to configure JsTestDriver to open the browser minimized. And you have to open the browsers because they run the JavaScript code.
You can try to create a virtual monitor and open the browser there.
Under linux you can use xvfb for this, for example
xvfb-run java -jar JsTestDriver-1.3.3d.jar --tests all --port 4224 --browser firefox chromium-browser --testOutput jsTests
should open the browser in a virtual window.