Not really. With Ambitions, you get an additional town that is built specifically to accommodate Ambition's new features. It's a whole new town, with different families and buildings. The original town, Sunset Valley, receives a couple of poorly-placed buildings to fit the new expansion. So you don't really play them separately, you just either use the new features and/or items, and that's it.
So to answer your question ("playing TS3 with Ambitions or could you in theory play the base game and then the expansion pack separately?"), yes. They features would be available to you irregardless, you can't just disable them, but you can choose to avoid them, if that's what you meant.