What’s “the resolution of a tab”?
The tab objects have a .height and .width property, which are “The height [/width] of the tab in pixels.” (whatever that includes).
The same goes for windows (but the documentation is more precise here).
Yes. just window.screen (in the background page) will give you an object with the dimensions of the primary display, of which .height and .width are useful.
I don’t think you can get more than that (at least not easily). Lats I checked devicePixelRatio was always 1 in the background page.
Edit: The tabs Window has a top and a left property, which are offsets from the top left corner of the primary display. Using these and the dimensions of the tab and the primary display, you can at least figure out whether a tab is displayed on the primary monitor.