@rachel92698 After you install the base game, you need to click on the tab that says "Expansion Packs" and download those as well, one by one. If you originally registered your packs, that's where they will be listed in your Origin library. And please remember that Origin didn't exist when TS3 was released, so there's no way it would know what you've bought unless you specifically input that information. It does share account details with the store, though, so if your packs are listed there, they should also be in Origin.
As fare as the contents of your game folder go, you can transfer most of the files from one computer to another. I used iCloud, which has a Windows-friendly app as well as the one that's built into mac OS. Your saves, saved sims and households, lots, and patterns can all be placed in your new game folder without issue. You do need to redownload all of your store content, but you don't have to pay for it again. The items will be in your account's purchase history.
I know that this isn't what you're asking, but you might want to be careful trying to run TS3 on a macbook air. Airs are not gaming computers—they have integrated graphics and lack the cooling strength necessary to protect themselves from overheating. Depending on your model, you might be able to run only the base game, or perhaps several of the less demanding packs. Pets and Seasons would likely place too much strain on your mac. And even if you got the game up and running, you'd still be much more limited than in Windows due to the way TS3 was ported into mac OS. But if you want a slightly more specific estimate of how your hardware might hold up, you can post your specs here. This is the relevant information:
https://answers.ea.com/t5/The-Sims-Mac/READ-FIRST-Sims-3-Mac-Help-Guides-and-Troubleshooting/m-p/4756624#M4