Windows 7 BSOD (Occurs while playing Video Games)

Hi,
whilst waiting for the diagnostics to finish I checked your DXDIAG and found this:
Drive: C:
Free Space: 2.6 GB
Total Space: 114.2 GB
File System: NTFS
Model: KINGSTON SV300S37A120G

This simply is not enough space for Windows to work correctly, you should always leave 20% free, the official figure is actually 10% but it's nice to have some wiggle room.

Bugchecks included 1A, memory management issue and 124, fatal hardware issue although by far bugcheck 1E appeared the most and a faulty device driver or system service might be responsible for this error. Hardware issues, such as BIOS incompatibilities, memory conflicts, and IRQ conflicts can also generate this error.

Bugcheck 124 is often caused by overheating and power issues. Please check your system is running within tolerances temperature wise:
HWiNFO64 is excellent for monitoring temps as it keeps track of a number of traces:
HWiNFO - Hardware Information, Analysis and Monitoring Tools

Please post the make and model of your PSU.

Bugcheck 1A can sometimes indicate the RAM is faulty. Windows does have a memory testing app but it can miss errors and the best app for the job is Memtest86.
If you open the link below you'll see you can run Memtest86 in two ways. You can either burn it to disk or install it onto a USB drive it's entirely up to you. You'll then need to enter the bios to change the boot order so you can boot from either the Disk or USB stick you have Memtest86 on.
You must test for at least 12 hours unless it becomes obvious there is a problem straight away.
Memtest86+ - Advanced Memory Diagnostic Tool

Try reseating the graphics card making sure any and all power cables are attached.

Please stress test the gpu by running Furmark. Monitor temperatures as well please:
FurMark: VGA Stress Test, Graphics Card and GPU Stability Test, Burn-in Test, OpenGL Benchmark and GPU Temperature | oZone3D.Net

Please post back with results or new dump files.