@gregsachs @Rand__ :
Thanks for your replies. I ran a VGA cable from the KVM itself to a VGA input on a video scaler/input switcher I use (4 HDMI, 2 VGA, DVI, Composite) that's hooked up to a portable 1080p HDMI monitor. I have one monitor and several devices (mostly HDMI) that need to output video, so this has been a good solution.
Last night, I did unplug the VGA cable after I made my post. I was too tired to do further experiments or post again (sorry), but:
It fixed all my problems. The noise disappeared, and I was actually getting the resolution I set as output in the Java KVM session window, instead of being locked to an unusable 640x480 with the video off center.
On closer inspection, the VGA cable I'm using is missing a pin. Apparently, that might be normal on some cables--Pin 9 is often missing.
Normally, that's not a problem, as that's supposed to be a ground pin. But in my case, it seems to have been an issue. The output was noisy (not properly grounded) and the unit was mad enough not to be able to set resolution correctly.
The more troubling alternative is that a pin actually broke off; I'll need to check the receptacle on the KVM to make sure nothing's stuck in there.
I need to do some further testing to confirm this was the actual issue, and probably get a new 15 pin VGA cable to test with, as well.
At least the web console is working as expected now.

If I had this to do over again, I'd get the unit fully configured before plugging in a display, so I could verify the HTML5 session output was correct. I'd have had to use the serial interface for that, as by default the LAN interfaces are effectively disabled (DHCP is off by default, so they can't pull addresses).
EDIT:
So, I was looking at bad information. Wikipedia is a better source for VGA cable specs:
VGA connector - Wikipedia . Pin 9
used to be for keying the connection.
It is now a 5V power pin for EDID EEPROM chips in some displays. Remember how the MPU was misidentifying its resolution and reporting as 640x480? Like its EDID was incorrect and it was defaulting to base VGA resolution? Sometimes, pin 13 and 15 play into this functionality as well, depending on the age of the monitior.
This KVM is only about 10-15 years old. It likely has the latest (last) VGA standard, which means it likely expects pin 9 to actually be the EDID PWR pin.
Definitely going to try with a proper cable.