Is there any way that my girlfriend and I can interact between our Sim characters/families? Is it possible to share neighborhoods online or play the same family change by change? We have both Origin accounts with Sims 3.
Due to the mechanics of story progression, the base Sims 3 game doesn't allow multiple controllable families at once in a single neighborhood.
This means, if you want your sims to interact, you have few options:
Play in the same neighborhood. Changing the active household whenever you want to "switch" will remove wishes, and puts the other characters at the whims of story progression. This may mean unwanted marriages, children, or moving out of the neighborhood entirely. Some of this can be mitigated by turning off story progression, but you'll still lose active wishes etc.
Use a mod. There are various mods around the web (Awesomemod is one, I believe) that work around the issues with the method mentioned above. Depending on the mod, you may even be able to get a "Sims 2" like feel, where time only passes for the active household. Recommending specific mods, however, is beyond the scope of this question and this site, so just know that it's an option.