Ok so this is my system

core2duo e6420
abit ab9pro socket775 mobo
eVGA 8800 GTS 320MB
4 sata HDDs, 3 on RAID as secondary HD, windows on one seperate sata HD
4GB corsair mem, 1 GB kingston, set to dual channel asynchronous setup, 3GB channel 1, 2GB channel 2.
Hardware is capable of handling all of this as is. Been running for nearly 4 years. Only about 58 cycles. (Only reset or turned off 58 times).

The other day, my gf was working on movie maker and went to publish her project when it stopped responding. As a favor to her, I let it stay until it would resume (knowing that 99.9% it would not resume).

I let it continue and started playing starcraft 2. About 5 minutes in I started noticing vertical lines, then the mouse started hanging. Immediately threw in a new battery, still nothing. It would work sporadically, but the computer was not frozen because I could alt-tab into windows where the video distortion was still apparent. 

As I alt-tabbed back into the game, I let my opponent know that my mouse was not working, and he said "neither is your internet, I was about to drop you". So, my ethernet was failing simultaneously, apparently.

I shut off the computer using the power button.

It restarts fine, goes into windows. About 3 minutes in, distortion appears again. mouse starts locking up. Computer then freezes. I shut it off.

I leave it that way for several hours.

I disconnect the power, open the side, make sure every is fine and nothing smells burnt. Upon putting it back on, I'm greeted by the high-pitch, low-pitch alarm that I had set to off in the bios. The alarm is from the cpu fan being below RPM, however, I have a fanless sink, so that is why it is set to off. 

Now, because this has also been happening recently along with power disruption (removal of cord), it makes me think that the mobo battery is out, because it is supposed to save that info even during power loss. (obviously)

Along with the alarm, I also get a long beep followed by 2 short ones. According to the internet, this is either a memory problem, or a video problem. Granted, because of the video distortion, I'm inclined to say it's the video card. My POST (indicated on the mobo) is hanging at 7F (one step away from FF, which is 'all clear'), and beeping on 2A.

(the alarm is fixed, i simply moved the case fan power source to CPUFAN. I was too lazy to do it before but I wasn't sure I was hearing the two small beeps with it blaring)

So I remove the video card----same problem.
I remove the ram----same problem.
I remove the pci cards----same problem.
I adjust the thing that resets the mobo----same problem
I remove and reseat the battery---same problem
I clean the video card (massive amounts of dust inside the compartment), reseat it, ----same problem.
I remove all RAM testing each one by themselves----same problem.

Here is a description of what's happening besides that;

No display, monitor reads 'no input'
All fans power up, including graphics card fan
USB is still working, repeated pressing on keys results in the clicking sound in the pc to let me know it has some effect.
I've tried holding insert while booting...nothing.
When I remove the power source and do an initial start, it hangs at C1 (memory POST code), then resets itself, makes it through C1, C3, beeps at 2A, 25, 26, 50-something, 75, hangs at 7F.
Once it reaches 7F it will continue that way for a long period of time. However, if I press delete a couple times, it seems to go through POST again, beeping again at 2A.

What in the hell is going on?

Memory problem?
Video problem?
Mobo problem?
Complete fry of the entire setup?

vertical lines always sounds like a video card error. However the rest not so much, so I guess your video card was too hot and thus has maybe damaged the mainboard.

I don't really know but I'd say the damaged part is one (at least one) of the following:
video card
mainboard
power supply

I'm gonna guess PSU or Motherboard as the problem seems so widespread and you've checked virtually everything else.

Can you try a different PSU? 

I would try video card first. Do you have access to someones spare card?

Edit: If it was a PSU undervoltage you would tend to get memory or HDD errors first, they are the canary in the coal mine so to speak for PSUs.

I posted on tomshardware (excellent site, long time user), and two other people were mentioning power supply issues.

Is this something that's really common with PSUs?

I'm totally at a loss since I know a lot less about power supplies than I do about the rest, which coincides with my inability to explain what is happening.

It's an 850W PSU that retailed for 140 on sale.

It cannot be underpowered. The only possibility would be it somehow became underpowered in a way that I don't really understand.

I was really hoping it was something I could just upgrade.

If I have to buy a new power supply, it's like buying a useless part IMO.

The problem is, the only other video card I have is AGP, and my mobo doesn't support it. Basically nothing out of my other PC will suffice. I believe I used the power supply from it previously once but didn't have enough wattage to handle the graphic card.

How about you're posting what for a PSU you got ?

It's not common when you have a good one, but if something happens and the PSU is at fault, there's a good chance that other parts of your PC will suffer as well.

exact name of the PSU please.

power and price doesn't really help much

Dunno how common, but I know my girlfriend and brother experienced a range of problems due to PSUs. With my girlfriend we tried everything before realising it was the PSU, including formatting the HDD (well, attempting to but the PSU made sure it screwed that up too!).

As Squilliam mentioned, HDD and memory problems tend to arise first with PSU problems. My brother also had problems with his DVD drive before it completely went.

The Wattage won't mean much if it's failing on you. My brother had a good PSU and it still died.