This may be a dumb question, but how does one get access to the fan bearing to lubricate it? Do you drip oil into the gap surrounding the hub, or is there a way to disassemble the fan and remove the blades?
As for the cold-start power-on, I noticed something about the startup circuit schematic:
Look at C9 and R30. When the Mac is first plugged in, C9 will be at 0 volts and then will begin to charge up to 5V through R30 and the 5V trickle supply. But it's a big capacitor and an even bigger resistor, 47 uF times 100K ohms is a time constant of 4.7 seconds. During the time that C9 is charging up, its input to the NAND gate will be low, forcing the NAND output to be high, which keeps Q2 turned off. What this means: for a period of a few seconds after the Mac is first plugged in, C9 prevents the Mac from turning on even if you press the power switch. I assume this is intentional, to prevent exactly the sort of spurious start-up that I'm seeing.
But my IIcx turns on within about 0.5 seconds of plugging it in or flipping the switch on the ATX supply. That should be impossible. So this points to a problem with C9, R30, or the connections between them and the NAND gate. But I've checked all those and they're OK.