screen freezes randomly without any specific warning.

Are you using the factory chipset? I'd take a look at the PCI controller driver and what Win7 wants for that video card.

Then if necessary try the EVGA latest.

Check that all WIndows Updates are installed after enabling Automatically get recommended drivers and updates for your hardware.

Installed all of the recent updates and still having an issue.
It has crashed a second time, this time while watching a YouTube video and talking with a friend on Skype.

Here's the BSOD info for that.

EDIT: Even though both of my hard drives passed inspections, I've heard a single click occasionally coming from (from what I assume) is the original hard drive that was inside the PC when I bought it. I never stored anything on that hard drive and I've only just had it inside my PC for "in case I need more room" purposes.

I've heard of "The Click of Death" but I had assumed that to be multiple clicks and not just one. But to be on the safe side, I've removed said hard drive and am now seeing how working with the newer one alone will work. I'll respond either IF there's a crash or in the next few days as a general update.

Blue Screen of Death (BSOD) Posting Instructions
When posting a new crash dump, include all System File Collection and health reports. They contain valuable information that we can use to find the cause of the crashes.
Most of the time just the dump files are not enough information.

These two crashes are the same as the first two, indicating CPU.

Code:

Probably caused by : AuthenticAMD
WHEA_UNCORRECTABLE_ERROR (124)
A fatal hardware error has occurred. Parameter 1 identifies the type of error
source that reported the error. Parameter 2 holds the address of the
WHEA_ERROR_RECORD structure that describes the error conditon.
Arguments:
Arg1: 0000000000000000, Machine Check Exception
Arg2: fffffa8007e31028, Address of the WHEA_ERROR_RECORD structure.
Arg3: 00000000bc800800, High order 32-bits of the MCi_STATUS value.
Arg4: 0000000000010137, Low order 32-bits of the MCi_STATUS value.
IMAGE_NAME:  AuthenticAMD
FAILURE_BUCKET_ID:  X64_0x124_AuthenticAMD_PROCESSOR_CACHE

OCCT graphs:
Most of the frequency graphs aren't very steady, which is likely caused by the inconsistent voltage.
The voltage graphs are not near as smooth as they should be, with the exception of the +3.3v graph.
The CPU Vcore is very bad, when your system is at idle it is extremely bad.

Do you have Cool'n'Quiet enabled?
Your CPU frequency doesn't change but the voltage does try to change.

I wouldn't put any new components in this system, they may get damaged by the voltage fluctuations.
From this system's history, I would say the old PSU has probably damaged the motherboard and/or CPU. Other components may be damaged as well.
The only way to test this is to install another motherboard in your system and see if it's stable.
Another option is to put your CPU in another known good system and see if the OCCT graphs look the same.
You can make sure all the drivers are up to date and the BIOS is updated, but I don't think this will help, won't hurt to try.

Another way to try and determine what component is responsible for the problem is to simplify the system.

Remove all non-essential components from the system, including USB connected devices and printers, leaving only the core components: PS, MB, CPU, Video Card, Monitor, Keyboard & mouse, and one hard drive. You could go even further and install only 1 stick of RAM in slot A1 to test.
When removing add on cards be sure to remove them from Device Manager first before physically removing them from the machine.
You can go further by removing the on board devices too, sound and LAN, first by removing from Device Manager, then by disabling them in BIOS.
The extent of this test is up to what you have the time and patience for.

It is also a good idea to perform a Clear CMOS once you have it down to the core system. This will also reset the BIOS Settings defaults and the Real Time Clock (RTC). This will ensure a "clean slate".
When performing the Clear CMOS take note of your SATA Controller setting in BIOS before. If it is set to AHCI mode you will want to re-enter BIOS and change that back before allowing Windows to start.
Leave all the other setting on default until satisfied with the test.

Now run the machine to test. With one stick of RAM it will be noticeably slower. Ignore the lack of performance (and functionality) for the test. Do the things you usually do when it freezes. You will not be able to test Skype or anything web related without LAN, but you can without sound.

It will either freeze again, which will prove that one of those core components is at fault, or it will not.
If it doesn't, then you want to add components back one at a time, testing again between each re-installation. Even add the additional RAM sticks back one at a time. If the freeze occurs after adding a component then you have your cause.

Let us know what you find.

Didn't know you replaced everything except CPU and ODD, would be nice to know these details from the start.

Follow TVeblen and Greg's suggestions, good idea to run another CPU stress test.
OCCT uses Linpack and the graphs give useful info.