i think you might require a different psu-- probably something like the m2, m4, or opus 360. part of the problem is that your current psu requires a constant 12v input to work correctly-- it is too dumb to know what to do with anything different. so while i believe that is half of the issue, i don't believe that it is the full issue-- i really don't think that any psu/pc combo should drop your battery voltage that low..
you might need to trouble shoot the wiring and make sure that it is all ok..
wiring: in my setup, the carpc and amp have separate power wires, though it really doesn't matter-- they all share a common point at the battery, and i have seen plenty of setups that run a single power wire to the back of the car that then splits to the carpc and amps...
btw, that cap is useless(some of oldsparks first posts covered this in depth), so i would normally start by not using one.
OS: i don't see any major advantage to x64 in-car unless there is a specific program that requires it.. in general x86 should be fine for most incar pc-ing