It isn't a molex 12v connector - it is the round centre pin type (the sort of thing you often find on consumer devices where you plug in the power adaptor). I suspect it is to provide power to the speakers that came with the PC.
No need to suspect - it is for powering the speakers, as I mention at the end of my post. The reference to molex connectors was only to indicate that some
PSUs provide a DC power outlet for powering other things, eg. HDDs.
The PSU cable supplied with the machin has a mains plug for the wall socket and a length of cable into what looks like a sealed junction box. Out of the junction box come two identical power cables with standard connectors. One for the monitor, one for the computer. I presume that rather than use a PSU with a standard power input socket and an output socket to daisy chain the monitor they are effectively providing the power to the monitor by simply splitting the mains lead.
no longer make PSUs with both a AC inlet and outlet because it was a bad idea in the first place. Actually the real reason is probably because they could not guarantee what some people would plug into them, eg. electric kettle.
If this is the case I am a bit nonplussed why a short car journey should kill the PSU altogether.
Without knowing the initial fault symptoms and your mention in your initial post that it failed to power on when you plugged it in, I only mentioned that it could
be the PSU.
Now that you have provided more info, the possibilities are, (in no particular order):
1) Intermittent short either from the motherboard to ground or between tracks/leads on the motherboard.
2) Fractured solder connection, (usually around a component lead), dry joint or cracked pcb trace.
3) Board/RAM/CPU not plugged in correctly, ie. not fully and securely inserted.
4) Switchmode PSUs will die, it's a fact that can't be changed and it is generally due to one thing only, High ESR (Effective Series Resistance) in the electrolytic capacitors. You can either spend the time repairing it, (as I've done on a couple), or try another ATX PSU.
5) Faulty CPU/RAM/HDD/etc.
6) You facing East-West on a North-South ley line
7) Practically anything.
The fact that it would POST without the DVD and then plugging it in caused it to not POST is suggestive of insufficient PSU power output at load.
Also, if the CPU HS/FAN was that filthy you can probably bet the the interior of the PSU is as bad if not worse, which would only serve to shorten it's life.
Personally, given the age of the machine, it's symptoms of intermittently failing to start, RAM errors, etc - the first thing I'd be trying is a known working ATX PSU.
Also, BIOS readings for voltages cannot be relied upon. The only correct way is a DMM while the PSU is under it's normal operating load.