I am still clear where your problem lies. You say the machine will not start/post, but you said you are getting into the BIOS. Which is it?

 
After having to restore Windows to factory settings, it started as a delayed post. I would press the power button and nothing would happen for about 10 minutes. I was still able to get into the BIOS from that. Eventually, it stopped posting completely.
 
I tried virtually *EVERY* possible solution known to man over the course of five days, which took up my entire time off days from work. I even tried a different power supply from a good computer. None of it worked. Tried everything suggested in this thread and every thread I made across various computer forums. Interestingly, when I put the power supply from the non-functioning Gateway into my Dell, the Dell actually booted up. Both of those power supplies were 300 watts, however. The ONE thing I haven't tried is a higher wattage power supply. I'd need to wait until I get paid on Friday to do that because all of the computer stores around me (no exaggeration) have closed down within the last 1-2 years. I live near the Northwestern Indiana border, so I may do some Googling for computer stores in that area.
 
It seems like the USB ports are getting power as my iLok will still light up. After trying everything, this is why I'm wondering if the CPU is dead. That is my top fear. I know I need to replace the motherboard regardless (I accidentally bent some pins on the CPU socket :( ) The thing that tells me that the CPU is NOT dead is the fact that it was posting originally, albeit a delayed post, before it stopped posting completely. Honestly, I'm not sure if that's typical of a failed CPU. The obvious culprit is the PSU, but given everything I said, I have my doubt that's the only problem. I believe the motherboard started to fail before it completely died and in the process, it took some components with it.
 
I just don't want to wind up ordering a mobo, a new power supply and have it turn out that the CPU is dead then I'm about $150. Part of me just wants to bypass that potential extreme frustration and just save up to get a new (custom) rig at the end of the month.

You can't swap it out I assume?
 
I know you don't want to hear this, but I've been in this position before. I got parts from different suppliers and they refused to swap them out stating incompatibilities with a display card I bought elsewhere, or vice versa. Even getting parts from the same supplier they would blame me for doing something incorrectly. Bare in mind I used to work for Samsung and I have built maybe 1000+ custom PC's in the past.
 
So I got fed up and stopped building my own PC's and ended up using off the shelf desktops, and I put on extended engineer next day warranties on them. That way it takes a phone call to get it fixed.

Do you have a system speaker attached?

Y: good
N: bugger!

I shall assume Y.  Otherwise, diagnosing whether it's the CPU, MOBO or RAM or 'X' can be a b1tch.  Refer to your MOBO manual as to the beep codes and their meaning, this is the best way of diagnosing the issue.  They look like this: http://www.cwc-group.com/casp.html

If you have >1 RAM stick installed, try booting with just one installed at a time, changing which one each time ofc.  This will also have the effect of re-seating the RAM, which is another possible problem.

You can test the PSU á la this video: https://www.youtube.com/watch?v=5FWXgQSokF4

There are q. a number of other things you can try (e.g. remove gfx card and just use the integrated gfx), but the beep codes are really the way to go.

A power supply with insufficient wattage can certainly jack things up.  I used to run with 500 watt supplies as a minimum, and about 3-4 years ago upped that to start using power supplies of 750 watts or higher, because I usually have a bunch of hard drives added (4-5 usually).
 
I made the switch to 750 watts or higher after my son's computer began having problems after adding some additional hard drives - and the power supply was 600 watts.  Swapping out the power supply to a 750 watt one instantly got rid of his issues.  For any computer I build for ME, I use a 1000 watt power supply.
 
If you can pick up a power supply in the 600-750 watt range, and that doesn't fix things, you should be able to box it back up and return it - and can ask, prior to buying the power supply.
 
Bob Bone
 

>I was installing a (legit) music production program from the DVD drive
>when Windows 10 somehow, weirdly glitched up and wouldn't read the disc. 
 
Have you tried disconnecting the DVD drive?
3 times in the last 10 years I've run into bootup problems (slow startup or complete failure) on various PCs caused by a dead or dying CD-Rom/DVD drive.