<<< Is there any way to run programs such as Microsoft Office from this USB hard disk rather than having to re install them on the new C: >>>
As things stand, 'No' because your new "C" drive with Windows in doesn't have the registry entries relating to the programs in the external drive. You would have to reinstall the programs, although you could then specify the external drive for the Program Files folder.
<<< If not, how do I un-install the programs from the USB drive to free up storage space? >>>
Delete the sub-folders in the Programs Folder. You will probably want to delete the Windows folder too because you cannot boot your original installation in the new hardware.