If you are having problems launching the game through the Origin client, try using the shortcut icon for the TS3 base game from inside the folder named "The Sims 3" inside Applications on your Mac's hard drive. Does that work any better?
Are you sure that your download and installation through Origin was OK? Is this a new problem you are having or has it happened for a while? If it is new, have you changed anything recently in your setup?
Also be aware that you may need to manually purge your RAM in order to get the game to run on your Macs. 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. 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.
Caspin
