As far as the socializing goes, I think you need to be firm. If Annie shows up at your house (seriously? that's pretty far out) you tell her through the door to go away, and then ignore her. Don't make excuses, tell her right out that you're not interested in socializing outside of work. She hasn't gotten subtle hints, so giving more won't work. And as someone pointed out upstream, if it looks like you guys have a social relationship, you could be seen as covering for her due to friendship, which could get you blamed if/when she causes another serious accident.
On the work front, I think you need to be more active. It sounds like Annie is dangerous. It's one thing to risk your own health and life, but if you don't push things eventually someone else under your supervision could be seriously injured. If they decide to make a fuss, you and your employer could get in serious trouble for knowingly retaining someone who is incompetent to the point of being hazardous to other employees.
One useful approach is to document. Every time there is an incident, you note the time, what happened, the potential (or actual) consequences of her incompetence, and the people who were there. When you talk to management, share that list. I'd bring up the fact that she shows up unannounced at your *home* as well, because that's more evidence that she doesn't understand even very basic interactions. Stress the damage she's caused so far, and the fact that you're worried about both the life and safety of the other employees, and the potential for serious injury and/or lawsuits. Bring in witnesses if needed.
And stop worrying about her sensitivity. This situation has gotten to the point where trying to be nice is likely to physically endanger you, your job, and your fellow employees. Trying to cover for her butterfly chasing is less an issue of being sensitive to an employee's mental state, and more like hiding a bus driver's alcoholism because you don't want him to get fired. The potential repercussions of shielding them are just too high.