Overall it might appear you are encountering some Corruption on your system. Several things might cause this but memory or a hard drive problem could be involved.
I found the thread below you might try to see if you can get the Windows Update problem resolved. One thing it does is rename the SoftwareDistribution folder, which is one step usually taken to clean up a Data problem.
Edit: You are getting this message over and over in your WindowsUpdate.log:
FATAL: Failed to initialize datastore, error = 0xC80003F3
Redirecting
You don't have to do a batch file, but up to you.
If you actually got a BSOD, if you can find the minidump file, attach here or in the BSOD forum...
I had the same idea just before I saw your post here. With all of the various symptoms and none of which really linking to each other I ran system diagnostics. Short DST was failed so I'm backing up all I can as it could go at any time.
However, that really doesn't explain the update services which were the reason for this post. Unless the Windows Update files are stored on a bad sector of the hard drive it doesn't make sense.
I ran the commands mentioned in the batch file listed above basically re-naming a folder and re-connected to update servers. Connection to both servers was successful. Windows update shows it has never received updates and is checking now so the Update problem is solved. I was also able to receive updates for MSE. [ATTACH type="full" align="right" alt="An image from 'Windows Update Service Is not Running / Possible Error with Startup'. 1. Commands to reset Windows Update components via Command Prompt.
2. Windows Update checking for updates with no prior checks or installs.
3. Notification showing 138 important updates ready to install, totaling 1,372.1 MB.
4. Successful completion of system file scan with no integrity violations found."]27259[/ATTACH][ATTACH type="full" align="right" alt="An image from 'Windows Update Service Is not Running / Possible Error with Startup'. 1. Commands to reset Windows Update components via Command Prompt.
2. Windows Update checking for updates with no prior checks or installs.
3. Notification showing 138 important updates ready to install, totaling 1,372.1 MB.
4. Successful completion of system file scan with no integrity violations found."]27260[/ATTACH] [ATTACH type="full" align="right" alt="An image from 'Windows Update Service Is not Running / Possible Error with Startup'. 1. Commands to reset Windows Update components via Command Prompt.
2. Windows Update checking for updates with no prior checks or installs.
3. Notification showing 138 important updates ready to install, totaling 1,372.1 MB.
4. Successful completion of system file scan with no integrity violations found."]27261[/ATTACH]
Running sfc /scannow produced no results.
[ATTACH type="full" align="right" alt="An image from 'Windows Update Service Is not Running / Possible Error with Startup'. 1. Commands to reset Windows Update components via Command Prompt.
2. Windows Update checking for updates with no prior checks or installs.
3. Notification showing 138 important updates ready to install, totaling 1,372.1 MB.
4. Successful completion of system file scan with no integrity violations found."]27256[/ATTACH]