Common BSOD or no boot-up

Hey, guys. New to this site. Decided I need help.

TLDR below.
Anyways, I bought my custom-built computer from a mom-and-pop store here in town for about $600. It had 2 sticks of ADATA RAM, 4 gigs total. Honestly, I don't know much about this computer. A friend of mine said he looked into the mobo I had and he said it was about a $400 mobo. Which means that only $200 was spent in everything else.
I got it home and it worked fine for a couple of months until I started getting BSODs. I didn't know what was going on and I was getting multiple error readings. "Page in Nonpaged Area", "IRQL not less or equal", etc. (something like that.) I sent the comp back to the store and asked them to find the problem and repair it. They said fine and after a week or so, I got my computer back. Of course, it was BSODing again. I called them up and the only thing they did was let it run for 2 days. They couldn't find a problem so they handed it back to me with a total of $80. I started messing with it. I reinstalled the motherboard CD and it started working fine. Of course, it didn't last long. BSODs everywhere! Same problems I was having before. I took out a stick of RAM and turned my computer on and everything looked fine, again. Until it BSODed again, lolz.

Now to the present:
I've ran Memtestx86 and concluded that both of my RAM sticks are in proper shape. Eventually, I could get my computer to turn on, but only to Safe Mode before it either crashed or BSODed. Even in Safe Mode, it would BSOD.

Randomly, my little brother decided to get on and he ran Startup Repair, somehow fixing the problem. My computer was working fine. Until I shut it down, in fear of burning something up for being on so long (it was on for a week or two, no restarts or anything). Now, my computer is giving me the same problem. It gets power, but does not boot up. I get 3 beeps, then a short pause before another beep. That's when I know it'll actually boot. It'll start up for a short time. I'm not sure what is going on. I unplugged the hard drive and RAM and it still wouldn't boot. Not sure if that's a mobo or power supply problem. Right now, it's turned off but the longer I keep it that way, the longer it'll boot up. I even resetted the CMOS but switching the jumpers but nothing changed. Then I went to the BIOS setup (whatever it's called) and before I could do anything, my computer let out this long beep and froze. It continued until I had to hold down the power button and shut it off.

TLDR; Power turns on but does not boot. Occasionally boots but commonly BSODs. Cannot turn off computer if I don't want it to restart this problem (if actually working). Unplugged RAM & HDD both separately and simultaneously, still no result. Wat do?

SPECS:
4 gigs of Adata RAM, 2 sticks each. 1 stick in, currently.
AM3 65W M26GT4-A Atheros AR-8113 10/100 Lan Chip for M26GT4-A
NVIDIA MCP61P Chipset
Not sure on hard drive.
Windows 7 64bit.

At the very least, I have a bunch of wallpapers, pictures, music, etc, that I would like to keep. I have an external hard drive I could put them on. It's about 200 gigs. (I totally forgot. It's late.) I can buy a computer at Wal-Mart or somewhere brand-new but I'd like to keep my stuff. Any tips, info, or hints on how to do that?

Much appreciated! Muchas gracias! Domo arigatou!
 
Those beeps are BIOS Code beeps and they tell us where the problem is. What BIOS is your system running (the motherboard manual should tell you; also it appears at the very first screens as you boot up.

Do you have any recent BSODs? Have you run a virus scan?
 
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