> you can't double-click to open a new tab, or middle-click to close a tab (and remove the close button at the end)

Added those features.
You can try pre-release from https://github.com/asamuzaK/sidebarTabs/tree/master/dest

> It doesn't show whether a tab is loaded or not.

Set `svg.context-properties.content.enabled` to `true` from `about:config`.

> it would be really cool if you could filter the list by container

Not planned. Out of purpose.