LR, does the HP have a sticker on the case with the product key on it? Does it match the product key shown on start>computer>right click "properties"?
Are you sure that the Product KEY gets shown? I thought the Product ID gets shown. I was unaware that the Product KEY is shown anywhere in Windows without using some third party software.
From the sounds of the problem I wouldn't be surprised if it was malware of some flavour causing it.