Added LG monitor and cannot get display icon to show

System>Library>Coreservices>Menu Extras - Double click on the Displays.Menu icon

Just like in the old (10.3?) days.

VGA cables are generally not designed for the high bandwidth required of large, digital displays, so it wouldn't surprise me if this is a cable issue that would be present in a variety of cables. I had similar issues with a KVM adapter. (If you're using a KVM adapter, then the KVM itself is probably the problem.)

One thing to try would be to lower the refresh rate on the monitor to the lowest setting the Mac will allow and see if that solves the problem.

Better would be to get the DVI cable working, but again the intermittent noise/snow points to a bandwidth limitation of the cable (or KVM adapter).