I just got Seasons on Amazon, however I already had the base game installed via disk. You need an Origin account, that uses the same login data as your Sims 3 account on the store site.
When you have your Origin account set up, if you don't already have one, you can redeem the code on the store site and then download them via Origin. They should appear under my games.
The different length in code is because the code redeeming option is for store content, like bonus items or worlds. You want to register a game. The option is right on top of "redeem a code". It can be a bit confusing, but the Amazon site should actually explain it. I think mine did, but I am in Germany and used Amazon.de. And I just checked the the US site, which does not explain this.
Definitely make sure, that your Origin account is set up first and uses the same login data.
ETA: I just checked and actually couldn't find this explanation for the base game. Maybe you do redeem the base game differently. It is possible to redeem a code directly on Origin. How to do that is described here: http://help.ea.com/en/article/how-to-redeem-an-origin-product-code/
ETA again: I also found this: http://help.ea.com/en/article/how-to-install-the-sims-3-through-origin/ explanation.
I am sorry, if I confused you further. Maybe you could check, if your Amazon download really didn't come with any instructions and maybe someone, who bought the same product can help you better. I found reviews mentioning both methods to claim the codes on the Amazon.com site for the starter pack, which is the one I assume you bought?
Is this the one: http://www.amazon.com/Sims-Starter-Pack-Online-Game/dp/B00CQOG8LI/ref=sr_1_2_title_0?s=videogames&ie=UTF8&qid=1411086878&sr=1-2&keywords=the+sims+3 ?
The third review actually gives the same instructions, I followed. They worked perfectly fine for me. Registering the EP on the store site made them show up in my Origin account under My Games and I could install it from there.
