All those issues were introduced in the Unity8/mir but most of them are being worked on if not already fixed
The keyboard showing in the external display has been closed just today I think.
So for now, you have to restart unity8/reboot after plugging in the external display for it work. Scaling is fine in the right side of the screen but if you want, you can set system scaling to 8 (1080p monitor) so that everything looks fine.