New, self-built PC keeps getting BSOD at random intervals.

VBAT is good then, it's a 3V battery and will still work down to around 2V.
If you use the stock Intel CPU cooler even a new fan won't help much. I have a Hyper 212 EVO and it works well for normal use and would be a lot better.

Yes, that's definitely possible.
Even though the CPU chip is pretty small and in the center of the CPU case, the case helps dissipate the heat.

If by fan you mean the CPU cooler, then I would recommend it. Just changing the fan won't help.
I've used the older version of the CPU cooler that Britton30 recommended and it is very good for about U$31 at Newegg in the US, should be at a reasonable price in your area.
COOLER MASTER Hyper 212 EVO

Get a new CPU cooler and thermal paste, be sure to clean off the old paste with Isopropyl alcohol before applying the new thermal paste and seating the new cooler.

New fan installed and I'm sure I've done everything correctly, but... The PC crashed again. No Blue screen this time as I switched it off before it could get that far. I've updated the SF Diagnostics, but I don't think that will add to anything. I'll run the CPU cooling program for half an hour again and upload the results to see if the fan is actually doing its job.

CPU Test Results:

What exactly do you mean by "New fan installed"?

Did you get a new fan or new CPU cooler?

The CPU core temps are better, acceptable for an Intel stock cooler.
Still concerned about the CPU Vcore fluctuation.
Have you ever overclocked the CPU?

No new dump files in your zip folder.
If it crashes again be sure to give it enough time to generate the dump file, we need this information.

From System Event Logs:

Code:

Event[20809]:
  Log Name: System
  Source: Microsoft-Windows-Kernel-Power
  Date: 2012-09-11T16:22:48.226
  Event ID: 89
  Task: N/A
  Level: Information
  Opcode: Info
  Keyword: N/A
  User: S-1-5-18
  User Name: NT AUTHORITY\SYSTEM
  Computer: Deadpool-PC
  Description: 
ACPI thermal zone ACPI\ThermalZone\TZ01 has been enumerated.             
_PSV = 372K             
_TC1 = 1             
_TC2 = 5             
_TSP = 1000ms             
_AC0 = 0K             
_AC1 = 0K             
_AC2 = 0K             
_AC3 = 0K             
_AC4 = 0K             
_AC5 = 0K             
_AC6 = 0K             
_AC7 = 0K             
_AC8 = 0K             
_AC9 = 0K             
_CRT = 372K             
_HOT = 0K             
_PSL - see event data.

This is still showing that the CPU is over heating.

App Event Logs:

Code:

Event[0]:
  Log Name: Application
  Source: NVIDIA OpenGL Driver
  Date: 2012-09-08T19:39:50.000
  Event ID: 2
  Task: N/A
  Level: Warning
  Opcode: N/A
  Keyword: Classic
  User: N/A
  User Name: N/A
  Computer: Deadpool-PC
  Description: 
The NVIDIA OpenGL driver has encountered an out of memory error. This application might behave inconsistently and fail.

Event[45027]:
  Log Name: Application
  Source: NVIDIA OpenGL Driver
  Date: 2012-09-08T19:39:56.000
  Event ID: 2
  Task: N/A
  Level: Warning
  Opcode: N/A
  Keyword: Classic
  User: N/A
  User Name: N/A
  Computer: Deadpool-PC
  Description: 
The NVIDIA OpenGL driver has encountered an out of memory error. This application might behave inconsistently and fail.

You had 45,027 of these on the 8th, you need to update your Nvidia driver, if you have the current driver or still are having issues install older drivers until you get one that works.
Be sure to use Driver Sweeper to remove all remnants of the old drivers, do this every time you install a new Nvidia driver.
What brand is your GTX 560?
Some of these cards are known to be unstable.

Your only uploaded dump file on 29 August:

Code:

STOP 0x0000007E: SYSTEM_THREAD_EXCEPTION_NOT_HANDLED    (go to top of page)
Usual causes:  Insufficient disk space, Device driver, Video card, BIOS, Breakpoint with no debugger attached, Hardware incompatibility, Faulty system service, Memory, 3rd party remote control,
IMAGE_NAME:  intelppm.sys - CPU driver
FAILURE_BUCKET_ID:  X64_0x7E_intelppm+114c

There may be other causes but this is pointing to the CPU, either from overheating or you may have a driver issue or CPU issue.

First, find a Nvidia driver that will work with your graphics card.
Second, run AS SSD benchmark on your SSD, post a snip of the finished run.

See if this helps with the thermal paste. There are many ways to do it, but this way works well.

Same dump file as in your last uploads, date 29 August.

Let us know your progress.