@SamuraiLady
You definitely need Origin for Sims 4 to run. I was explaining about Origin not being needed for Sims 3 because you referenced it in your posts. It has no effect on getting Sims 3 to run as the files are completely separate, unlike Sims 4 which requires Origin for authentication. The only issue Origin might cause is a corrupted download.
Unfortunately Sims 3 is not supported in Mojave, it hasn't been officially supported since OS X Mountain Lion and this is an issue for some players. There is no expectation that the game should run on unsupported Macs as it hasn't been updated since January 2014 (which was a cursory update anyway).
If I were you I would:
Completely and cleanly uninstall Sims 3 including any shortcuts to the game
Run the Origin reset tool - https://help.ea.com/en-gb/help/origin/origin/how-to-reset-origin-on-mac/ to get rid of any downloaded .dmg files from previous downloads if present
Empty trash and restart your Mac
Start up Origin and download and install the base game
Patch the base game by running the 1.67 manual patch, not through the launcher
Start up the game and check it is working. If so carry on with the next steps
Download each pack and install it in release date order. Once installed open the Launcher and download any further patches it prompts for. When you install the problematic packs - Ambitions, Late Night, Fast Lane, Outdoor Living - make sure to add them to the Launcher. When you then install subsequent packs you must follow Corey's instructions to add those packs to the pack that the game launches. It is tempting to skip this step, to install all packs then copy the folders into the launching pack, but I advise against this because you want to launch the game after installing each pack to make sure it is working. In order to launch it successfully you must copy the folders over for Ambitions, Late Night, Fast Lane and Outdoor Living
