How to fix the error codes 80072EFE and 800736B3

The 80072EFE error is simply a bad internet connection - although that can be due either to physical problems or software interferences by malware, firewalls, etc.

The 800736B3 error is somewhat different.

Please follow the Windows Update Posting Instructions and post the requested data.

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remove the two or three oldest CBS.persist.cab files - they aren't likely to have anything relevant in them

You seem to have removed the latest CAB file - which has the SFC data in it.
Please post that when you can.

The older one has the following errors....

Code:

2013-05-18 21:41:07, Info                  CSI    000000b6 Performing 1 operations; 1 are not lock/unlock and follow:
  Install (5): flags: 0 tlc: [9984c73279f3a80e9759ec258e502cfd, Version = 6.1.7601.18126, pA = PROCESSOR_ARCHITECTURE_AMD64 (9), Culture neutral, VersionScope = 1 nonSxS, PublicKeyToken = {l:8 b:31bf3856ad364e35}, Type neutral, TypeName neutral, PublicKey neutral]) ref: ( flgs: 00000000 guid: {d16d444c-56d8-11d5-882d-0080c847b195} name: [l:154{77}]"Package_1_for_KB2830290~31bf3856ad364e35~amd64~~6.1.1.2.2830290-2_neutral_GDR" ncdata: [l:2{1}]"2") thumbprint: (null)
2013-05-18 21:41:07, Error                 CSI    000000b7@2013/5/18:11:41:07.100 (F) d:\win7sp1_gdr\base\wcp\componentstore\storelayout.cpp(8701): Error STATUS_SXS_ASSEMBLY_NOT_FOUND originated in function ComponentStore::CRawStoreLayout::QueryComponentUnstagedFiles expression: (null)
[gle=0x80004005]
2013-05-18 21:41:07, Error                 CSI    000000b8 (F) STATUS_SXS_ASSEMBLY_NOT_FOUND #254678# from CCSDirectTransaction::PerformChangeAnalysis(...)[gle=0xd0150004]

Your CheckSUR report shows many 'errors' - all of which are in fact a problem with IE10 coding, and standard for such installs.
What is unusual is the number of languages which you have apparently installed in IE10 - and I see that there are actually a couple of 'errors' missing - the Arabic and Hebrew Hyphenation MUM files (if they actually exist - I haven't attempted to install those languages to find out)

Please open an Elevated Command Prompt and run the following commands....

Code:

 
REG LOAD HKLM\COMPONENTS C:\Windows\System32\config\COMPONENTS
REG QUERY "HKEY_LOCAL_MACHINE\COMPONENTS\CanonicalData\Deployments\9984c73279f..c258e502cfd_31bf3856ad364e35_6.1.7601.18126_88ff8ec1a727dd62"
REG QUERY "HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Component Based Servicing\ComponentDetect\amd64_microsoft-windows-lddmcore_31bf3856ad364e35_0.0.0.0_none_2bc129cfe41472e8"
REG UNLOAD HKLM\COMPONENTS
 
.

Here are some instructions to make life easier :)
1) To open an Elevated Command Prompt Window (the ECP window), click on Start, All Programs, Accessories – then right-click on Command Prompt, and select Run as Administrator. Accept the UAC prompt.
2) To run the commands easier, highlight the block of commands, and right-click on the highlight – select Copy. In the CP Window, click on the black/white icon at top left – select Paste. The commands will run but may not complete the last command, so hit the Enter Key once.
3) To copy the results... click on the Black/White icon in the top left, and select Edit... 'Select All', and hit the Enter key - then use Ctrl+V or r-click+Paste to paste it into your response.