Can I install the Windows XP (Home/media center) that comes with a DELL PC on a different PC instead?

It is not possible to install XP using the Dell CD on another PC that is not a Dell. They do use a feature that detects wether or not it is a Dell. However since your new Dell comes with a liscense for windows xp, you could use that key code when installing XP on your older machine, but you will need to use a Windows XP install disk that is not manufacturer-specific, for instance borrow a CD from someone who purched the copy of Windows XP by itself.

Yes, use the windows cd key that is on the side of your computer case. When you try to activate it, windows will say it couldn't activate it online. You will have to call the number, which will hook you up to a machine which will ask you if you changed your motherboard - say yes. Then you just give the machine the numbers you see on the screen, and the machine will give you over the phone a code to enter to validate it. This will invalidate the installation on the dell though. It's not just designed for dells. A windows XP cd key is a windows XP cd key.

It's possible, some of the OEM disk look at something on the computer that tells them it's a dell or manufactor and the cd will only install on that one... but if you can get around it or have another cd it would be legal to do so.