Somehow literally 'copy' the OS across?
No, don't do that with WinXP. Because of device drivers especially AHCI, you won't get happy.
Download the files I need to install a new copy? (and then activate it using my key)
Yes you can, but be sure to get a good, clean copy/distribution, there are some out there. You need to know, if your key and your media is OEM or Retail, because WinXP keys make a difference in that. For example if your certificate is from IBM, DELL or HP, you need the OEM media. Also make sure to install the correct edition Home or Pro.
If you choose the right setup media, activation will complete successfully.