Aurora R10, will not boot without use of power supply button

If it is under warranty you should contact Dell, this sounds like a hardware issue possibly with the PSU or led connector/power button wiring for example.

An update to this issue: The computer will now not properly go into sleep mode. When I attempt to put the computer into sleep mode, it loses power. To boot again as previously described, one must push the button on the power supply and the power button on the front of the case.

I'm going to attempt running all of the alienware updates to see if this helps the issue.

yep; their systems are down, still standing by.

The oddest thing, the computer just randomly came back on while I was sitting at the other machine. I can't remember if this was from shutdown or from a previous attempt to put it in sleep mode.

Another followup: 

I've now unplugged all devices except mouse and keyboard and attempted to restore the computer to factory standards. 

This has also NOT fixed the issue.

It's starting to seem more and more like a faulty PSU or possible MB.

The power button issue is a known issue that a number of people have reported on.  The workaround sometimes fixes it, not always.  The randomly turning on is a new twist but might be related to the power button not working without a jumpstart.

This is a long shot, but when I have my phone plugged into the USB power charging port on my R10, it wants to keep booting from the phone and restarts multiple times during the boot sequence.

I am wondering if anything plugged into one of your USB ports is causing an issue for you?

Can you try unplugging all USB cables from your Alienware, and see if the problem still exists afterwards?

Having to turn off the power supply indicates to me an issue with the standby voltage, where you have to remove the standby voltage to get the motherboard back into a normal state. I have seen this before with oddball USB issues.

That's what I did above when I did a factory reset, but to no avail.

Thanks for suggesting however.

Is it still in OEM configuration, or have you upgraded anything like RAM? 

OEM except additional SSD. Dell is replacing PSU though dispatch has been delayed.