I decided to test the ignition switch (IS). The wiring diagram indicates that Splice Block 100 (SB100) feeds wiring harness pins 5 and 7 on the B(+) side of the circuit, and outputs through 1, 2, 3, 4, and 6 on the B(-) side. I did not remove the harness from the IS.

I performed continuity tests to the IS.
SB100 to SB102: Pos 0 (OFF) no continuity. Pos 1 (ACC), Pos 2 (ON), Pos 3 (START) has continuity.
SB100 to ICM 3D connector: Pos 0, Pos 1 has continuity. Pos 2, Pos 3, no continuity.
SB100 to SB101: Pos 0, Pos 1 no continuity, Pos 2, Pos 3 has continuity.
SB100 to Fuse 33: Pos 0, Pos 1, Pos 3, no continuity. Pos 2 has continuity.

I did not test continuity from SB100 to C200. Current flows from SB100, through the IS when it is in Pos 3 (START), and on to connector C200 (to PCM, Pulse Module, and Clutch Interlock Switch). I assume this is fine since the car cranks up and starts.

Basically since I was not able to get access to the back of the IS, I tested continuity from the splice block that precedes the wiring harness, to the next connection in the circuit. By connecting a multimeter I was able to turn the key through the different positions and get continuity that appears to match what the schematic shows. The IS seems to be working fine.

Replacing the ICM did not alleviate symptoms. Since the symptoms seem to stem from operating the headlamp switch and brake pedal I suspect the issue lies down the line from this module.

I believe testing the Central Locking Pump / Security System Module (CLPSSM) would be a good idea next. This is connected to the ICM, and I've been having problems with the alarm siren going off on its own. The CLPSSM is connected to the Body Control Module (BCM) via the CAN B BUS. It seems conceivable that if this module fails it can really do some screwy things. The BCM does not directly control anything that is displaying a symptom though.