Add automated tests

[WofWca]

At least some basic, smoke-test level stuff. This should make things a little easier for new contributors.

Unit tests would be easier to implements, but I think end-to-end / Selenium are more valuable. Both are good, of course. Not sure if Selenium can work with extensions.

[WofWca]

Here's Google's guide on how to do this. They suggest Selenium, Puppeteer, WebDriverIO:
https://developer.chrome.com/docs/extensions/how-to/test/end-to-end-testing

Also for reference, this extension https://github.com/Authenticator-Extension/Authenticator has thorough E2E test coverage.