holsinthehouse01 You can delete the Origin Games folder since there's nothing in there related to a different game. (I didn't want to break any other pre-EA App game installs of yours.) The EADM entry is likely unnecessary but also shouldn't be a problem, or at least I haven't seen it be a problem. Everything under Electronic Arts looks to be related to the EA App; if you wanted to flush this, I'd say use Revo to remove the App instead, since the App is finicky and there are a lot of other hidden files that you wouldn't want to try to delete by hand.
One other thing to check, which hasn't traditionally been an issue for Sims 3 but definitely is for other EA games, is a hidden License folder. Click Windows key-R and enter this in the run box:
%ProgramData%\Electronic Arts\EA Services
You can take a look at the License folder inside, or just move it to the desktop to force the EA App to create a new one. Normally, that would happen when you tried to launch a game that needed anything in this folder—you wouldn't need to reinstall. But as I said, I don't know whether this comes into play with Sims 3 at all, and if so, how things behave.
If none of this helps, I will install Sims 3 through the EA App on my testing laptop and see what I get, then compare it to your registry entries. One question though: have you installed any packs (this time around), or is it just the base game? There used to be one entry per pack, but maybe that's changed since I last looked at an EA App install.