Sorry for the delay in responding - it's been a little hectic and I've been working on my new system (selfish, I know
As I understand it, you can't physically connect your front panel (power/lights/etc) connector to the appropriate pins on the motherboard. I've not seen a connector like the one you have, but it's probably a proprietary setup (my older AW Predator case had the usual individual connectors).
If this is a "Physical" connection problem, I doubt it indicates a short or electrical problem in your motherboard - although it is a bit confusing, as you indicate you've tried plugging the connector in various ways.
I suggested the extender as an easy way to see if you could connect the Power Switch ... that will only work if the connectors are small enough to fit into the receptacle that is holding the pins in the case connector. If it will fit, and you can isolate which pins are actually the Power Button (try tracing them from the back of the Power button, if you can get into that part of the case) - then you can plug them into the power switch pins on the motherboard (check your diagram). You won't hurt anything if you get them the wrong way ... just turn off the system and plug them in the other way.
If this works, then I'd suggest your best bet will be to rewire the Alienware front connector. It looks like you can remove the individual wires (still in their pins) from the plastic. Take a paperclip and carefully press down on the pin in the little square openings (just above your thumb in the picture), and see if you can push the pin out with the wire intact.
If this works, then you can simply get some two pin connectors and re-insert the appropriate pins in each ... you'll need to trace them to the source.
The reset button is used when you want to do a reboot without actually turning everything off ...
It might help if you post a picture of your desktop, and maybe we can find someone with a similar machine who can help.
Note - this may not be the easiest way to get things going, just the only logical way I can think to troubleshoot it without actually seeing the connections.