This file does not have a program associated with it for performing this act

That's strange. That file should repair the registry. However, you can try manually repairing the registry. Before doing so, try setting a System Restore point. Because of your problems with .exe files I'm not sure if you will be able to so I'll show you another way to backup your registry once inside the registry editor.

Press Start>Run and type cmd in the box and press OK.
At the command prompt type cd c:\windows and press return.
Type copy regedit.exe regedit.com and press return.
type regedit.cm and press return. You should now be in the registry editor. On the menu bar click file and select export. Save the registry as registry.bak.