Windows 7 - Computer not using all of its available ram.

I have 6gb DDR3, more than 75% of the time my PC wont allow me to use all 6. When I do get all 6 running (from restarting PC over and over until it kicks in) it usually runs for about 30 mins until it crashes, without a blue-screen or any message. And then when I turn it back on, it sometimes keeps failing POST, beeping 4 times - indicating a RAM read/write failure.

When it's using 4gb however I rarely ever crash.I've checked the BIOS, it usually shows 4gb installed.

I've had this problem for more than a year and I thought I had fixed the problem a while back but I was wrong, and now it's doing it worst than ever. Also, i'm not computer savvy so bear with me.

(Thumbnails below are SIW, task manager and resource monitor.)

Sounds like either some Ram slots or Ram modules are defective.
Use this to test what is good and what isn't :
RAM - Test with Memtest86+

-DG

I don't have a USB stick or any dvd-r blank discs. But i have Daemon Tools, could i use it somehow with that?

nope you need any bootable device. besides the resident driver of Daemon Tools is a known cause for BSODs.
Do yourself a favor and remove it with this: DuplexSecure - FAQ

If you still need a Disc emulating program, here are some that don't rely on resident drivers:

Freeware MagicISO Virtual CD/DVD-ROM(MagicDisc) Overview
SlySoft Virtual CloneDrive
Gizmo Central -- Enliven your PC

-DG