Windows 7 Service Pack 1 problem (error 8024200D)

It looks to me as if your problem lies in the SoftwareDistribution folder, rather than anywhere else.

Code:

 Line 635615: 2014-06-08 14:44:53, Error                 CBS    Exec: Failed to pre-stage package: WUClient-SelfUpdate-Core-AdmComp~31bf3856ad364e35~amd64~pl-PL~7.6.7600.256, file: rdpcorekmts.dll, source: \\?\C:\Windows\SoftwareDistribution\Download\19b53483b79dbd4efbf89b4fc7b270d4\amd64_microsoft-windows-t..extensions-binaries_31bf3856ad364e35_6.1.7601.21907_none_0264a4fb0f9b0e60\rdpcorekmts.dll, sandbox: (null) [HRESULT = 0x800f081f - CBS_E_SOURCE_MISSING]
 Line 635619: 2014-06-08 14:44:53, Error                 CBS    Failed to pre- stage package: Package_for_KB976932~31bf3856ad364e35~amd64~~6.1.1.17514 [HRESULT = 0x800f081f - CBS_E_SOURCE_MISSING]

Open an Elevated Command Prompt
Run the following commands

NET STOP WUAUSERV
REN C:\WINDOWS\SoftwareDistribution SDOLD
NET START WUAUSERV

reboot
then wait 10 minutes, and open Windows Updates, and run a Check for Updates.

Install only the oldest update found - and reboot once installed
then go back to Windows updates and install the others.

Try using the standalone installer -
Download Windows 7 and Windows Server 2008 R2 Service Pack 1 (KB976932) from Official Microsoft Download Center

You'll need the windows6.1-KB976932-X64.exe file

First, run Disk Cleanup, and tick all the boxes - reboot at least twice.
Delete the C:\Windows\SDOLD folder that we created earlier
Empty your Recycle Bin.
How much space is there now?
50GB for a Windows 7 partition is rather on the tight side - I don't suppose you have spare room in another partition, that we can steal?

How much room do you have now?
Using the standalone x64 installer requires up to 7.4 GB free space.

You may be able to free up some more space by switching off Hibernation and deleting the hiberfil.sys file

Made Disk Cleanup and deleted folder - have 3,5 GB. I have second partition with 80 GB unused, but how can I transfer free space to system partition?

#EDIT: Okay, made some free space with transfering second profile files to another partition (the owner isn't using it now anyways). But now I got different error: "A system error prevented the service pack from installing." It says to use Check for System Update Readiness. Error code is 80070002 (ERROR_FILE_NOT_FOUND). What's wrong now? ;/

I suspect that moving a profile wasn't the right way to go about freeing up disk space. You should move the data back before doing anything else.

The only real way to move free space from one partition to another is fairly complex, and rather fraught with potential dangers.
It involves using a 3rd-party partition manager to shrink the secondary partition, then move it to the end of the drive, and then expand the system partition (assuming that partition is at the front of the secondary one).
If all goes well, then no problem - but a minor error can result in a need to completely reformat and reinstall.

I have successfully talked someone through this in a forum - but also had a failure. Both cases were some time ago, and my preferred too is no longer available from the developer, and may not work properly nowadays anyhow, after the changes in Windows.

...

You seem to need to move/delete 4GB from the system drive to be able to install SP1 properly - this is going to be a fairly tall order.

Let's have a good look at the system...
Please follow the Blue Screen of Death (BSOD) Posting Instructions and post the resulting file - it should give me enough to be able to start the cleanup process without endangering the system integrity.

Interesting -
Your HOSTS file appears to be corrupt - it's probably worth deleting and rebuilding it, as it may be inhibiting proper access to things other than the internet.

I'll work out some proper instructions for it and come back.