Hi
Using the digital download versions of the game from Origin is certainly a way to avoid problems with the disc authorisation error. However you need to fully and cleanly uninstall the disc-requiring version of the game before you install from Origin, otherwise it may keep asking you for a disc. See the more detailed instructions below. Once you have the game installed, consider playing in windowed mode, as you can then use "force quit" to get out of the game if it freezes. Also see the general Mac tips below, including purging RAM, as this can really help the game to run well on Mac.
If you have further problems let us know.
Caspin
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.
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.
Tips for Mac simmers
Unfortunately a lot of TS3 players encounter problems when playing the game on their Mac. These can include crashing and freezing. A lot of these issues can be avoided or at least improved by the following measures.
Firstly, manually reset your RAM while playing. See BluebellFlora's guide on how to do this: http://bluebellflora.com/the-sims-3-on-a-mac/resetting-ram-during-gameplay/
I restart my computer before I play TS3. Close down other things that are running in the background. The memory used as seen in Activity Monitor is at about 3GB after the restart. If I open the Launcher it goes up to around 11 GB (that's all I've done - just opened the Launcher!). At this point I reset the RAM. Then I hit play and get through the opening sequence. I then press cmd-return to put the game in windowed mode from time to time while playing (and always before saving or using CAS). That way I can keep an eye on the memory and reset RAM if necessary. Being in windowed mode also allows you to close the game using force quit if it freezes up - it's better to crash the game than your whole computer.
Second tip is to save frequently. I save at least once per sim day. Use the option to Save or Save As, rather than the combined Save and Quit.
Make sure you are patched up to date. The 1.63 patch included fixes for Mac simmers that did actually improve the stability of the game on our operating systems, so it is well worth updating to at least that level. The current patch is 1.67.
The game runs better if you switch off the memories system and disable the in-game Store and social features.
For general advice, information and tips on how to play TS3 on a Mac, read BluebellFlora's guide here: http://bluebellflora.com/how-to-play-the-sims-3-on-a-mac/