Pin 21 is the input from the remote board. Pin 12 is the degauss output. Given that you're having random issues with the OSD showing up uninvited, it sounds like it's an issue with that IC receiving what it interprets to be commands from the remote board (i.e. voltages other than 5V), rather than the degauss doing weird things. Remember that the exit button is also the degauss button if no OSD is up, so that makes some sense.
Yes, take your meter and connect black to ground and red to that pin (21). Be careful not to bridge pins. The 10 or 20VDC (whichever you have) setting should be correct for this.
To check resistors, use the resistance (ohms) setting on your meter. Apply one probe to one lead of the resistor and the other to the other side. Which probe goes where doesn't matter. Tracing out the remote board (the schematic is incomplete, notice the extra zener diode and R407; yes there's two R407s...), it looks like you should be able to test them on the board (unplug it from the chassis after power down) presuming the switches themselves are good. If you get weird readings (i.e. something not expected), remove the resistor from the board and test out of circuit.
R406 should read 100 ohms, R408 should be 470, R409 should be 1k (1000), and R410 should be 2.4k (2400). Use an appropriate scale. Anything in the ballpark (the resistors are 5% tolernace) is fine.
The diode (D401) is actually a zener and can't be easily tested with some multimeters. I can't read the full part number to determine specs on it, either, and it doesn't appear in the schematic. I *think* it's a 1N5231, which would be a 5.1V zener. It looks like it's being used as a clamp. This could cause the failure you're seeing. If you have a diode check setting, use that. Apply the red probe to the cathode (side with the black band) and the black probe to the anode (other side). Should read about 5V, but with the low current multimeters use for test this reading can be erroneous, and not all multimeters will test diodes to this voltage. You can also check it the other way: should read about 0.5-0.7V. Again, low test currents can cause this to be low. Given the circuit, you shouldn't have to remove the diode from the board for this test, but if you are having trouble, I'd suggest doing it anyway. I don't have an authoritative schematic to go by as the one in service manual I have appears incomplete for the remote board.