BSOD while gaming and web browsing

Okay, I looked into it and found that my copy of Windows wasn't activated yet. I went ahead and did the steps to activate Windows, no problem. I restarted after activating Windows, just in-case and proceeded to install the update you linked. I am going on vacation tomorrow, and I don't want to leave the computer running MemTest for the duration of that time, so I'll have to get back to you on doing MemTest.

These are the new results after activating Windows and installing the update, using the Microsoft tool you linked to me.

Code:

Diagnostic Report (1.9.0027.0):
-----------------------------------------
Windows Validation Data-->

Validation Code: 0
Cached Online Validation Code: N/A, hr = 0xc004f012
Windows Product Key: *****-*****-Q82BJ-RJKHY-6G4X7
Windows Product Key Hash: 10j0Qsfa69jEMVyz3Trb5YI3AN4=
Windows Product ID: 00359-OEM-8802073-01633
Windows Product ID Type: 3
Windows License Type: OEM System Builder
Windows OS version: 6.1.7601.2.00010300.1.0.003
ID: {2896A703-AEA4-43E2-81D7-0F20746A0C11}(3)
Is Admin: Yes
TestCab: 0x0
LegitcheckControl ActiveX: N/A, hr = 0x80070002
Signed By: N/A, hr = 0x80070002
Product Name: Windows 7 Home Premium
Architecture: 0x00000009
Build lab: 7601.win7sp1_rtm.101119-1850
TTS Error: 
Validation Diagnostic: 
Resolution Status: N/A

Vista WgaER Data-->
ThreatID(s): N/A, hr = 0x80070002
Version: N/A, hr = 0x80070002

Windows XP Notifications Data-->
Cached Result: N/A, hr = 0x80070002
File Exists: No
Version: N/A, hr = 0x80070002
WgaTray.exe Signed By: N/A, hr = 0x80070002
WgaLogon.dll Signed By: N/A, hr = 0x80070002

OGA Notifications Data-->
Cached Result: N/A, hr = 0x80070002
Version: N/A, hr = 0x80070002
OGAExec.exe Signed By: N/A, hr = 0x80070002
OGAAddin.dll Signed By: N/A, hr = 0x80070002

OGA Data-->
Office Status: 109 N/A
OGA Version: N/A, 0x80070002
Signed By: N/A, hr = 0x80070002
Office Diagnostics: 025D1FF3-364-80041010_025D1FF3-229-80041010_025D1FF3-230-1_025D1FF3-517-80040154_025D1FF3-237-80040154_025D1FF3-238-2_025D1FF3-244-80070002_025D1FF3-258-3

Browser Data-->
Proxy settings: N/A
User Agent: Mozilla/4.0 (compatible; MSIE 8.0; Win32)
Default Browser: C:\Program Files (x86)\Mozilla Firefox\firefox.exe
Download signed ActiveX controls: Prompt
Download unsigned ActiveX controls: Disabled
Run ActiveX controls and plug-ins: Allowed
Initialize and script ActiveX controls not marked as safe: Disabled
Allow scripting of Internet Explorer Webbrowser control: Disabled
Active scripting: Allowed
Script ActiveX controls marked as safe for scripting: Allowed

File Scan Data-->

Other data-->
Office Details: <GenuineResults><MachineData><UGUID>{2896A703-AEA4-43E2-81D7-0F20746A0C11}</UGUID><Version>1.9.0027.0</Version><OS>6.1.7601.2.00010300.1.0.003</OS><Architecture>x64</Architecture><PKey>*****-*****-*****-*****-6G4X7</PKey><PID>00359-OEM-8802073-01633</PID><PIDType>3</PIDType><SID>S-1-5-21-975808724-1595229991-1731456076</SID><SYSTEM><Manufacturer>MSI</Manufacturer><Model>MS-7751</Model></SYSTEM><BIOS><Manufacturer>American Megatrends Inc.</Manufacturer><Version>V10.5</Version><SMBIOSVersion major="2" minor="7"/><Date>20120427000000.000000+000</Date></BIOS><HWID>6E7A0F00018400FE</HWID><UserLCID>0409</UserLCID><SystemLCID>0409</SystemLCID><TimeZone>Pacific Standard Time(GMT-08:00)</TimeZone><iJoin>0</iJoin><SBID><stat>3</stat><msppid></msppid><name></name><model></model></SBID><OEM/><GANotification/></MachineData><Software><Office><Result>109</Result><Products/><Applications/></Office></Software></GenuineResults>  

Spsys.log Content: 0x80070002

Licensing Data-->
Software licensing service version: 6.1.7601.17514

Name: Windows(R) 7, HomePremium edition
Description: Windows Operating System - Windows(R) 7, OEM_COA_NSLP channel
Activation ID: 9f83d90f-a151-4665-ae69-30b3f63ec659
Application ID: 55c92734-d682-4d71-983e-d6ec3f16059f
Extended PID: 00359-00176-020-701633-02-1033-7601.0000-3622012
Installation ID: 017393206333418974700686981181957764198995546250867273
Processor Certificate URL: http://go.microsoft.com/fwlink/?LinkID=88338
Machine Certificate URL: http://go.microsoft.com/fwlink/?LinkID=88339
Use License URL: http://go.microsoft.com/fwlink/?LinkID=88341
Product Key Certificate URL: http://go.microsoft.com/fwlink/?LinkID=88340
Partial Product Key: 6G4X7
License Status: Licensed
Remaining Windows rearm count: 3
Trusted time: 12/28/2012 10:40:19 PM

Windows Activation Technologies-->
HrOffline: 0x00000000
HrOnline: N/A
HealthStatus: 0x0000000000000000
Event Time Stamp: N/A
ActiveX: Registered, Version: 7.1.7600.16395
Admin Service: Registered, Version: 7.1.7600.16395
HealthStatus Bitmask Output:


HWID Data-->
HWID Hash Current: MAAAAAAABAABAAEAAAACAAAAAQABAAEAln1uMuCBEsI6MjCoWrSEKmI9vEMSri5z

OEM Activation 1.0 Data-->
N/A

OEM Activation 2.0 Data-->
BIOS valid for OA 2.0: yes, but no SLIC table
Windows marker version: N/A
OEMID and OEMTableID Consistent: N/A
BIOS Information: 
  ACPI Table Name    OEMID Value    OEMTableID Value
  APIC            ALASKA        A M I
  FACP            ALASKA        A M I
  HPET            ALASKA        A M I
  MCFG            ALASKA        A M I
  ASF!            INTEL          HCG
  SSDT            SataRe        SataTabl
  SSDT            SataRe        SataTabl
  SSDT            SataRe        SataTabl

Okay, I defaulted my motherboard and CPU overclock and almost have finished 7 passes of MemTest. Yesterday, I played CS:GO for a bit (albeit with my 4.6ghz overclock) without any problems. However, I think I should play a game more demanding than that for seeing if my system is stable.

Should I go ahead and install my Windows updates? What do you think I should do from here?

Edit: Okay, I've done 8 passes of MemTest with 0 errors.

Yes go for updates keep restarting until there is no more, this is where most updated drivers come from on 7
With regards to OC leave it off until this is sorted for all we know that could have been your problem

Personally i would do them all

Its small memory dump (minidump) you want to set it to
Dump Files - Configure Windows to Create on BSOD
Option 2 Step 2