Thanks for the superfast reply - I'll attempt to be faster myself.
Like you say with your laptop, I also have iTunes 12 and it works, so there must be something different with your desktop. A full uninstall/reinstall like you tried would have been what I'd try next. I'm not sure what else to suggest. You could try it again, and double check everything gets removed from the iTunes/iTunesKeys folders.
If you search for iTunes and regserver or part of that error message you can find other people who have had the same problem. I'm not aware of any solution other than the /regserver option or reinstalling.
I'm actually not even sure what the /regserver option does, if it is an external dll file or if it's something with the iTunes.exe itself. I originally found that solution from other people who had that problem with other iTunes add-ons.