Windows XP prompts you for a product key only once, and that is during installation. If you're getting prompted for a key AFTER installation, then your key is either blacklisted or already in use on another computer.
To answer your second question... no. If a retail version was installed on the laptop, you could remove it from the laptop and install it on the desktop... but you cannot use the same key on multiple computers. The days of 95/98/ME/2000 keys are over.
If the copy of Windows XP came with the laptop (OEM copy of Windows), then it cannot be removed from the laptop and installed on the desktop. (Well, at least according to MS's license agreement). If you purchased the Windows XP after you bought the laptop and it is a retail copy, then you are allowed to uninstall it from the laptop and install it on the desktop.
If it is an OEM copy, it may not install on the desktop anyway. Many OEM manufacturers (Dell, HP, etc) check the motherboard's BIOS to ensure that it's being installed on the particular type of machine it was made for... it won't install on a computer from a different manufacturer.
It is possible to buy a license, and it's usually cheaper than purchasing the CD... but since you have built a new computer, why not purchase an OEM copy of Windows? I don't think the price difference will be all that huge, and then you'll have a CD of your own.