Cannot Complete Installing Updates, or Check for New Updates

Ouch!
That does sound as if it's stalled
The following article may help Solution to Windows 7 Installation Hang at 62% or 72% during Upgrade from Vista
It's written for a Vista>Win7 upgrade, but should still apply for a repair install

I can barely believe it, but I've somehow regained my desktop. Interrupting the stalled installation resulted in an endless loop of Windows 7 setup screens that reported the error and recommended a restart, but a simple adjustment in the regedit persuaded the system to "prepare my desktop for the first time," as if the install had succeeded. Once at the desktop, it took a repair of the .NET framework to be able to be able to activate Windows, but I seem to have an operational machine again. I'll try doing updates and report back with the results.

UPDATE: Checking for updates has been successful so far. In the meantime, one of the final steps in the repair install walkthrough Tom provided was to use Disk Cleanup to remove the C:\Windows.old, C:\$INPLACE.~TR and C:\WINDOWS.~Q folders. When I ran Disk Cleanup, it didn't detect any leftover installation files to delete (presumably because of the way the installation wound up finishing), so C:\$INPLACE.~TR and C:\WINDOWS.~Q are still there (C:\Windows.old doesn't exist), and are accompanied by C:\$UPGRADE.~OS and C:\$WINDOWS.~LS. Is it safe for me to delete those folders manually?

There shouldn't be a problem in manually deleting those folders/files