There is a difference between registering you game and redeem codes for stuff. Use the codes for the links provided for redeeming but use your game SN number to register your game on TheSims3.com
You can register your game on the Origin site but since they are not fully sync'ed up with your TheSims3.com profile data the game will not show up on your My Page at least right away. Its best to register on TheSims3.com
If your SN will not work to register your game (which it will not if you registered on Origin first) then you need to contact Origin Help to help you fix that. They may have a typo in the code they are expecting or some other problem they have to clear.
I had a problem with pets when trying to register my game and it kept telling me the SN was invalid (that's when I learned about not registering on Origin first if I wanted the game to show up on TheSims3.com site right away). They had to fix something and then the game registered.
Unfortunately, when it comes to SN numbers (product unlock keys) there is no workaround or anything that you can do to fix it if it does not work - you need help from the tech guys who have the only access to fix something like that
