Yeah, James, you're best with going for an entirely new system. WoW is a big memory hog, but just adding more RAM isn't going to make it go much faster for a game like this with an old processor. The video card shouldn't make much difference if you're running it on all the lowest settings, but a new system comes with one, so what the hell.
You won't have any problems installing it like Volsh said. Your CD key is attached to your account, not your copy of WoW. You could go to someone else's computer, install WoW and play from there if you wanted and it would work fine.
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