Windows failed to install updates

Hmmm - very surprising...

Open Event Viewer
click on the Windows logs entry in the left pane to expand it.
Now click on the Application entry - wait while it loads.
Click on 'File' in the menu bar and select Save...
Save the file as Appevt.evtx
Repeat for the System log
then zip both, and upload them.

You appear to have a failure in the AMD Overdrive service. - not usually a critical problem, but worth investigating.

You also appear to have a problem (much more serious) with what seems to be your own user profile.

Log Name: Application
Source: Microsoft-Windows-Search
Date: 13/08/2014 20:46:35
Event ID: 3036
Task Category: Gatherer
Level: Warning
Keywords: Classic
User: N/A
Computer: Slade-PC
Description:
The content source <csc://{S-1-5-21-3497225080-4047220842-1123543599-1000}/> cannot be accessed.
Context: Application, SystemIndex Catalog
Details:
The object was not found. (HRESULT : 0x80041201) (0x80041201)
Event Xml:
<Event xmlns="http://schemas.microsoft.com/win/2004/08/events/event">
<System>
<Provider Name="Microsoft-Windows-Search" Guid="{CA4E628D-8567-4896-AB6B-835B221F373F}" EventSourceName="Windows Search Service" />
<EventID Qualifiers="32768">3036</EventID>
<Version>0</Version>
<Level>3</Level>
<Task>3</Task>
<Opcode>0</Opcode>
<Keywords>0x80000000000000</Keywords>
<TimeCreated SystemTime="2014-08-13T19:46:35.000000000Z" />
<EventRecordID>36064</EventRecordID>
<Correlation />
<Execution ProcessID="0" ThreadID="0" />
<Channel>Application</Channel>
<Computer>Slade-PC</Computer>
<Security />
</System>
<EventData>
<Data Name="ExtraInfo">
Context: Application, SystemIndex Catalog
Details:
The object was not found. (HRESULT : 0x80041201) (0x80041201)
</Data>
<Data Name="URL">csc://{S-1-5-21-3497225080-4047220842-1123543599-1000}/</Data>
</EventData>
</Event>

You may also have a problem with the Shadow Copy service...

Volume Shadow Copy Service error: Unexpected error calling routine CoCreateInstance. hr = 0x80070422, The service cannot be started, either because it is disabled or because it has no enabled devices associated with it.

I suspect that this last is the critical one.
The 422 error here usually means that the service has been disabled.
Please run the following commands, and post the results.

SC QC VSS
SC QUERYEX VSS


post the results.

You messed up the second command
SC QUERYEX VSS

That looks fine.

Since it's been some time since my request - let's have a new look at the event logs - please upload new copies, following the instructions in post 31 again.

There is a driver problem that appears to be from your AMD Graphics (Catalyst) card -

Code:

The AODDriver4.3 service failed to start due to the following error: 
The system cannot find the file specified.

I suggest uninstalling the drivers completely, and then installing a new set.
How-To Uninstall AMD Catalyst Drivers From A Windows Based System


You also have a number of errors associated with Windows Search - try clearing the Index, and letting it rebuild.
Click on Start, in the Search box, type 'Indexing Options' wait until the entry appears in the list, and click on it.
Click on the Advanced button
In the Index Settings tab, click on the 'Delete and rebuild Index' Rebuild button - OK out, then leave the machine alone for a good while - reboot, and post a new set of logs.