My first thought is get your own place, perhaps nearby, but don't live with the mom. When it comes down to it, your #1 priority is taking care of your baby, and you don't trust her in this regard. Perhaps it's time to address all of the issues you listed here less delicately with your boyfriend AND his mom.
On the other hand, perhaps you can't afford a place on your own, and perhaps you'll be depending on the mom for childcare assistance. Assuming you will be working, full time childcare is ridiculously expensive, so having someone around to help you out is invaluable. Of course, she could still help you out with childcare if she lives nearby.
If you decide to live with her, lay out some ground rules. For example, it would be better if she rents a room from you, as opposed to it being HER house. Then, she can be as big a slob as she wants in her space, but must respect the common areas. The reality is that her habits are not going to change, and keeping the house clean will most likely fall to you and your boyfriend. And, don't be shy about not letting her drive with the baby. If any of these topics are so touchy that you're unable to even discuss them, then that's even stronger evidence that you shouldn't live with her.
One final point, and that is to consider the power dynamic of your family. You didn't mention anything about the relationship between your boyfriend and his mom. 21 and still living at home sets off a few alarm bells. I have a hunch that their relationship is rife with codependency and unhealthy emotional attachment, but you didn't directly say that (though you did allude to never being able to get away from her once you go down this path). How well do you fit into all of this? Do they gang up on you? Is she overbearing with your boyfriend? With you? Perhaps this stuff is all my imagination, but my gut says this is not a good situation. You should not live with her if you can possibly avoid it, nor should you allow yourself to get bullied into something you're not comfortable with.