Clean uninstall/reinstall instructions
When you uninstall TS3 games on your Mac using the uninstaller, the game leaves behind stray files and isn't completely removed from your computer. These files may conflict with future installations. It is best to completely remove them. BluebellFlora's website provides a guide on how to do this:
http://bluebellflora.com/the-sims-3-on-a-mac/how-to-do-a-clean-uninstall-and-reinstall-on-a-mac/
After cleanly uninstalling, install the base game and patch it up to date. If you cannot update via the Launcher, download and apply the latest cumulative manual patch (called the superpatch) for Mac. Be sure to choose the right file - there are various 'regional' files for disc-users and one designated region 7 for Origin-users.
Check that your game works properly. Then install any EPs and SPs that you own, in order of release date.
To download and install games through Origin
Download and install the Origin client for Mac and make sure it is up-to-date. If you already have Origin on your Mac but are having problems with downloading or installing games, delete the client and download a new copy (this sometimes fixes issues with downloading). The latest version can be downloaded from here: http://www.dm.origin.com/mac/download
Additional advice on how to download games (thank you Tremayne for this tip):
When downloading from Origin, do NOT use the first install icon. Move your cursor over the game cover and click on the information (letter i) and then click on the install button from there. For some reason, downloads from Origin frequently become corrupted. Using the i reduces the chance of your download being corrupted. If you have already been unfortunate and have a corrupted download, then it is important to remove all traces of this before trying again, by doing a thorough, clean uninstallation first.
Are you running anti-virus software or any clean-up applications? If so, disable these, as they might be deleting files that Origin needs. A fast, stable internet connection will also help you.