If you can't figure out what email account you used back then, you'll need to contact EA customer support to get it sorted out. If you get a message stating that the code has already been registered, it's possible you added it to your account a long time ago. The two share information, so the game will be added to your account in both.
The same product code on the back of the insert in your CD case will allow you to add TS3 to your Origin account, or you can create an account at the Sims 3 store () and register the code there. Yes, you can download the game from Origin for free, provided you have registered your disc copy or can do so now.Information, Guides and Announcements for the EA app.