Thanks. I tried the below tests with the motherboard still physically in the magnesium frame and that frame within the "base cover" (bottom plastic case) so that I could ensure the fan is mounted correctly, but otherwise with only the items you mention electrically attached.
Nothing ever shows up on the external monitor for Test-1
Nothing ever shows up on the LCD for Test-2. Separately, I actually connected the LCD to the mobo of that other working T410 that I have and it works there just fine, so I don't think there is any issue with the actual backlight or the cable connection on the back of the LCD.
I also tested the RAM in the other working T410 and it is fine. Keyboard and charger are known to be ok. Fan spins (temporarily). CMOS battery doesn't seem to be required but I also haven't tried voltage testing it.
So here are the possibilities that I can think of:
1) The eBay replacement mobo that I got is not operational (this is my suspicion)
2) I attached the CPU to the mobo incorrectly
3) Something about the way the mobo is physically mounted into the frame/case/heat sink is causing a problem