Windows Update Failure (error 80072EE2)

Why not simply turn off the 'clean up desktop' option?
It's simply a registry switch - usually accessed with GPEDIT in the higher versions of Windows, but directly accessible using regedit.
However, the 'approved' way to turn it off in Home Premium and lower is to disable the Microsoft\Windows\Diagnosis task in the Task Scheduler - which has very little other effect on the system so far as I can make out.

Great - SFC is now clear.
You should be able to install any updates that come along now - do a manual Check for Updates, and see if anything new is found, and see if they install if so.

this may be your Network Controller getting in the way?
Run SFC and CheckSUR again, and I'll take another look and the files generated.

Just CheckSUR.log and CBS.log (zipped) please.

The SFC scan shows the following problems...

Code:

 Line 4222: 2014-06-18 01:12:38, Info                  CSI    000003c5 [SR] Repairing 1 components
 Line 4223: 2014-06-18 01:12:38, Info                  CSI    000003c6 [SR] Beginning Verify and Repair transaction
 Line 4226: 2014-06-18 01:12:38, Info                  CSI    000003c8 [SR] Cannot repair member file [l:50{25}]"TS_UnusedDesktopIcons.ps1" of Microsoft-Windows-MaintenanceDiagnostic, Version = 6.1.7600.16385, pA = PROCESSOR_ARCHITECTURE_AMD64 (9), Culture neutral, VersionScope = 1 nonSxS, PublicKeyToken = {l:8 b:31bf3856ad364e35}, Type neutral, TypeName neutral, PublicKey neutral in the store, hash mismatch
 Line 4229: 2014-06-18 01:12:38, Info                  CSI    000003ca [SR] Cannot repair member file [l:44{22}]"TS_BrokenShortcuts.ps1" of Microsoft-Windows-MaintenanceDiagnostic, Version = 6.1.7600.16385, pA = PROCESSOR_ARCHITECTURE_AMD64 (9), Culture neutral, VersionScope = 1 nonSxS, PublicKeyToken = {l:8 b:31bf3856ad364e35}, Type neutral, TypeName neutral, PublicKey neutral in the store, hash mismatch

I have no idea how that happened?

I'll post a fix protocol as soon as I can.