I backed up Windows 7 from one computer to an external hard drive. If I plug this in to a Vista machine and restore it from that, will it have windows 7 on it? I'm very skeptical but I want to know something about it before I try anything. In theory it should work, right?
^ Correct. If you did a full system backup, you could restore that from within the Windows 7 install disc by booting from it (there's an option there to restore a system backup). It's not going to work, however, if the specs vary at all. It might, if the motherboards are identical, but then again...
"Keep in mind that a windows 7 key can only be on one computer at a time unless it volume licensed."
This is a false statement as these PC's are from HP/Compaq and they contain OEM license which belong to the machines. You cannot copy move or restore an OS from one OEM machine to another without violating EULA license.