Random thought for one board:
I have a Willamette (180nm) Pentium 4, but it's a Socket 478 (normally these are 423 chips) - if I put it into a "modern" 478 board, the thing will behave as you described, and absolutely nothing happens. Is that potentially the case here?
Regarding the 423 board I have less ideas - does it need RD-RAM and is it missing C-RIMMs?
Other generic ideas: are any jumpers set wrong or not set at all? How are the CMOS batteries? Is a CMOS clear jumper engaged? (Some boards will refuse to start in that situation)
As far as booting with no CPU - I've never experienced a board to properly start-up without a CPU, at least that I can remember. I'm guessing there isn't an "all in one" rule of thumb for that though - some may work fine, some may not.