Testing done properly in Chrome extension

We are developing a Chrome extension that in it's core tracks state of the browser (windows + tabs) and updates UI accordingly.

We have assembled a long list of manual testing scenarios. This in it's essence looks something like this:

1. Change settings to: Close tabs when saving window = true
2. Open three tabs with "https://one.com", "https://two.com", "https://three.com"
3. Click Extension button (trigger "Save tabs and close window" action)
4. Test if:
   a) Window is closed
   b) Saved tabs are visible in our UI (one of the tabs)

What would be the best approach to automate these scenarios (they are plentiful, and manual testing is boring and prone to error).

The main problem is how to combine manipulation of the browser itself (opening tabs, closing tabs, closing windows) with testing of it's state (did the window really close) while also testing the UI (did the tabs appear in the DOM) and IndexedDB (if visual clues are not present in UI).

Any feedback would be really appreciated!

P.S.: For those interested - the project is Tablerone (https://tabler.one).