How do I remove the light approx. 1 px border of the unselected artboard (on the right)?

It's a display anomaly/bug. I've seen this for years at certain display percentages, usually around 1600% and above, and it depends on what monitor you are looking at it on. It tends to show up on high-pitch monitors (like an iMac Retina screen). It's actually the background artboard display that's off; the objects are fine. If you zoom in to a percentage where you see the "line", switch to outline mode and you will see the object properly lines up with the edge of the artboard.

I remember seeing this reported on the bugs list before, but obviously nothing has happened to fix it.

It just shows an unselected artboard. Why do you want to remove it? Does it print, export?

Your right path is not exactly aligned with the right artboard in size and/or position..

Please show us two more screenshots:
(change rulers to unit pixels)

1.  upper left corner - the left path is marked and zoomed in to the maximum
2. upper right corner - the right path marked and zoomed in to the maximum

Please check beforehand whether the sizes and positions of your two artboards are really exactly the same size and on integer positions.