First of all, I would turn all of your settings on low if you haven't already, but right now I smell corruption of a file. Do you have any expansion packs?
There's also a chance that a file in the main part of the games coding got corrupted, which could call for a reinstallation of the series. That was actually my first thought - you might end up having to uninstall/reinstall everything, as I'm under the impression that you've never had these issues before (or they aren't a common part of your game, which would lead to a hardware issue) so they must have developed. Since a lot of the issues carried over when you moved your family to a new save I'm inclined to believe it's likely deeper than just a corrupted save file alone... there's one way to test this:
Start a brand new file, play with brand new sims. If you have no issues, then your family is corrupted. If you have issues then it's likely that a file within your Sim's coding got corrupted or out of place - the only way of fixing that truly is by an uninstall/reinstall.
I don't own a Mac, but I've heard a factory reset is possible with them. Rename your The Sims 3 folder (if you notice it's not just the one family that is corrupt, but all the saves) and launch the game - don't worry about adding anything just yet. Open a save and mess around with the base stuff you have (no store/CC/extra saves) and see if that fixes it, if it doesn't then I'm afraid to say the only thing that can/might fix it would be a completely uninstall and reinstall. Please let me know if any of this helps! My fingers are crossed that you don't need to do any brutal uninstalls/reinstalls! Also, remember to back up any saves/downloads/preferences you have from your folder! If the Factory Reset works, don't delete the old folder until you're sure that you have everything you need out of it!