Alienware keeps randomly shutting down!!

Those are common symptoms of a video card issue, specially that buzzing sound.
The Video card is usually in charged of the audio as well.

Are you able to record the temperatures while you have the computer in idle?

If so. Can you also try getting some temps while you're running a game within Window-mode to see if it is the Video card or the CPU that's crashing the computer.

These are the temps I got. I don't know if they will help much. I'm going to have to put them on here as two seperate posts, it says the files are too big.

Ok, it says the files are too big themselves. I'll just tell you them. Idle temps are: 52, 52, 52,52 (degrees Celsius). Game temps are: 66, 65, 66, 71. I'm sorry I don't know how to show you them through the post.

Run this benchmark - (Once the program is running and you see the animation, press F9 to begin the test).
Report back to me with the results.

Well, I tried your benchmark test and the computer shut down before it could even finish. I could try running it again and turning the fan speed up so it doesn't overheat as much. I wanted to run it by you first.

Do you mind submitting a screenshot of your Device Manager, expanded only on the System Devices category?
Maybe there's a conflicting driver there.

Thanks

You seem to have 4 temps at once, this doesn't look like video card temps, but core temps of your CPU. Try using GPU-Z and send your vid card temps with standard control fans, not manual fan control. If you can't get all your temps from GPU-Z, try HwInfo and post your temps from that for your vid card. If it dies during your benchmark and your 4 temps posted are really your GPU, its not overheating, but having 4 temps at once sounds like your CPU core temps...

Also, your fans might be clean and all, but not running top speed from dust or just age, post your fan RPM's when gaming, and a good heat to them, and post RPM speeds of the fans as well, could be just the fans are old and need to be loved/replaced.

Had the same problem.  Pretty sure it is overheating.  It's a pretty tight case.  My works fine now,  The solution was to permanently remove the cover, unseat the hard drive and move it away from the motherboard.  It is now sitting on the floor wide open  on its wide back cover, LOL.   But no more crashes!!!   Good luck.

BTW ran all the tests, tried increasing fan speed, blah, blah,blah.... tried everything but ice cubes.  

I can't do that with my ccomputer, since i use it for school a lot. I have to move it around from classroom to classroom, as well as from my dorm and my family's house. So carrying it with the hard drive out and all those parts exposed probably isn't the best choice for me. I'm glad you were able to fix yours though.

Am I seeing that right? In the 2nd doc (fans rpm normally.docx), the GPU1 Current and lowest RPM speeds are 0? If that's the case, you found your problem. The fan is old and sticking at low voltage. Even GPU2 current at 400RPM seems slow. My fans at low temp and idle are at least 1100RPM. If under warranty, ask for new fans for GPU's and CPU. If feeling adventurous, repaste with some Arctic Silver 5 :)

If that's true about my fans, then why does my comp still shut down even though I have the Manuel fan control on so the rpms for both GPUs are around 2000? Please explain. I just wanna make sure before I try to get the new fans.

Do you mind sending me your service tag via PM, so I can take a closer on your system and it's components.
Hopefully I can find what would be the best solution in this case.