Ok, update: I'vb chatted with the customer support for 40 minutes now, I have downloaded origin, linked my ea account to my steam account and they have adeded the steam game in my origin-library after I send them proof that I bought the game from my account. When you now log in to the sims3-website with your ea-account, then the game is automatically registered and you don't need a cd-key anymore.
My advice for people with the same problem: First link your ea-ccount to steam (log out and then in the log in window on the bottom is a steam symbol.) Then contact them and let them add the game to your origin account. Saves them time.
@JuliaMarasha Are you going to play through Steam or Origin? If Steam, I'd be interested to know whether you can download any items from the Sims 3 Store. Now that your Origin account owns the game, you have access to the Store: just sign in with the same email and password you use for Origin.
Even if you don't want to buy any store content, you could try installing Riverview, which is free:
https://store.thesims3.com/riverview.html?categoryId=12642
But it's encrypted, like all other store content, which is why I'm curious whether it would work on your setup.
