Windows 7 SP1 fails install, error 0x800f081f

It looks like this is the cause of your problems...

Code:

Checking Component Store
(f)	CSI Missing Winning Component Key	0x00000000	x86_policy.8.0.microsoft.vc80.crt_1fc8b3b9a1e18e3b_8.0.50727.1801_none_516953ad0f4d16c4		

Summary:
Seconds executed: 269
 Found 1 errors
  CSI Missing Winning Component Key Total count: 1

This could be a very difficult one to get right, as it's one of the ones for Visual C++ and these updates are a bit of a law unto themselves
I would suggest attempting to uninstall all versions of Visual C 2005 - it should be in your installed Programs listing (leave any other Visual C installs alone such as 2008/2012 etc)
succeed or fail, then run a new CheckSUR scan and post the new log.

OK - I need to have a look at your COMPONENTS registry hive

Please copy the C:\Windows\System32\config\COMPONENTS file (no extension) to your desktop and then compress it - upload the compressed file to your favoured fileshare site (preferably Dropbox or OneDrive) and post a link.

OK...
Open an Elevated Command Prompt, and run the following commands

Code:

REG LOAD HKLM\COMPONENTS C:\Windows\System32\config\COMPONENTS
REG DELETE HKLM\COMPONENTS\DerivedData\Components\x86_policy.8.0.microsoft.vc80.crt_1fc8b3b9a1e18e3b_8.0.50727.1801_none_516953ad0f4d16c4
 
.

then reboot and run a new CheckSUR scan - post the results
Then try the update again.

CheckSUR is happy, at least!

Unfortunately, now that the woods have thinned, I can see the trees...

YOu appear to have a variant on a common problem....

Code:

2015-08-21 16:33:16, Info                  CBS    Exec: Staging Package: Microsoft-Windows-RemoteFX-VM-Setup-LanguagePack~31bf3856ad364e35~amd64~en-US~6.1.7601.17514, Update: INF_ts_wpdmtp, PinDeployment: amd64_synth3dvsc.inf-languagepack_31bf3856ad364e35_6.1.7601.17514_en-us_4ec14685497da0d0
2015-08-21 16:33:25, Info                  CBS    Calling client to resolve source, cannot find file 'TsUsbGD.sys' at path: \\?\C:\Windows\Servicing\Packages\amd64_tsgenericusbdriver.inf_31bf3856ad364e35_6.1.7601.17514_none_9872c8452ac8f816\TsUsbGD.sys
2015-08-21 16:33:25, Error                 CBS    Exec: Failed to pre-stage package: Package_for_KB976933~31bf3856ad364e35~amd64~cs-CZ~6.1.7601.17514, file: TsUsbGD.sys, source: \\?\C:\Windows\Servicing\Packages\amd64_tsgenericusbdriver.inf_31bf3856ad364e35_6.1.7601.17514_none_9872c8452ac8f816\TsUsbGD.sys, sandbox: (null) [HRESULT = 0x800f081f - CBS_E_SOURCE_MISSING]
2015-08-21 16:33:25, Info                  CBS    Failed to gather all required files. [HRESULT = 0x800f081f - CBS_E_SOURCE_MISSING]
2015-08-21 16:33:25, Info                  CBS    Failed to gather all missing files for package: Package_for_KB976932~31bf3856ad364e35~amd64~~6.1.1.17514 [HRESULT = 0x800f081f - CBS_E_SOURCE_MISSING]
2015-08-21 16:33:25, Info                  CSI    0000000e@2015/8/21:06:33:25.415 CSI Transaction @0x36cccb0 destroyed

More commonly I see a French language file being demanded, rather than a Czech version- but the prognosis is the same.
As far as I know, the only sure solution to this problem is a repair install