Games crashing on Aurora R3

Hey guys, so for a long time now (the last year or so) I've been battling with my computer to run the games I'm trying to play. (Borderlands 2, Arma 2: Combined Operations, Dota 2, Batman: Arkham City, Elder Scrolls: Skyrim, The Witcher 2) and a few other less known games. Everytime I play any of these games, only the applications of the games will crash to desktop. It can be on launch, it can be within 5 - 10 minutes of playing. I've tried everything from reinstalling Windows 7 cleanly, to replacing the motherboard, harddrive, video card, RAM, power supply (with a much higher wattage/better brand) tried it on multiple different monitors and tried with just plugging it into the wall instead of a surge protector. Ran all the tests like memtest, and made sure the computer wasn't overheating but everything is fine. All the latest drivers are installed 100%, and that's just the tip of the iceberg with what I've tried.

I've fought with Alienware tech support on fixing the computer, but they never came close. Tried the same things over and over (reinstalled Windows multiple times, updated drivers, same old same old) and then after a while just charged my computer as fixed and moved on. To date I'm fighting that charge and they also said they were going to not charge me for it, and reimburse me after they did, but again...never happened. Anyway, I then brought it to Best Buy to see if they could fix it, but alas they ended up not being able to fix it and I'm working on my refund of the tech support for that as well. Lesson learned about tech supports, I suppose. 

I'm about ready to just trash this computer and build an entirely new one, because not only do my games crash but other applications as well. Google Chrome is the most functional of my browsers and it still gives me a lot of trouble with crashing. Openoffice also crashes...So on and so forth. Most are a KERNELBASE.dll error, not sure if that's relevant. 

Short summary: Games and other applications are crashing, tried literally everything that I, and anyone else I've talked to can think of and still nothing.

What's strange is I can play some games, like Killing Floor or Just Cause 2 fine. No crashes, everything is good. Yet I can't for the life of me get any of my other games to work. Can someone help me? Also, sorry for the wall of text. It's been a rough time with this. Any help would be much, MUCH appreciated.

Alienware Aurora-R3
AMD Radeon 6950 2GB video card
intel i-5 2400 processor
Windows 7 64-bit
16 gigs of RAM
1TB HDD