You can copy it over. I had it on an external HDD for a time and it worked fine on any computer.
Strange. I would have said no without trying it. There are definitely some registry entries for it, which is usually the reason for installation programs. Maybe the client makes them if they don't already exist.
Though I can't imagine that they use COM, which requires registry entries of a most heinous nature, which is another reason for installation programs.
The registry entries for WoW seem only to hold information pertaining to where the client is located in the file structure, for the patch executables I imagine.