Just my opinion--
You are thinking WAY too much about being nice. Women are trained from early childhood in "how to be nice." This puts women at a unique risk for being abused and taken advantage of, because we are taught to be nice and sweet. Seriously, women die of being nice and giving the benefit of the doubt every day, it makes us easy prey for crazies or abusers.
This is a very simple situation. Someone only vaguely acquainted with you showed up at your door with a weapon, threatening to kill your pet. You later find out that this person stalked (yes, stalked, and I don't think that is too harsh a word) your son to find out where he lived, with the specific intent of making this threat.
Either the person is crazy and needs to be evaluated by professionals and possibly placed in an assisted living facility or this person is a serious threat to your family and neighbors.
I think you are discounting your neighbor's ability to cause harm because you can't imagine that a little old lady would do such a thing. If you want countless examples of little old ladies who get up to all kinds of dickens, feel free to volunteer at your local old folks home with the Alzheimer's patients. They are capable of hurting and abusing people, no problem. Just try to tell them that you checked and they do NOT have a doctor's appointment today, or that you're not taking them to IHOP again because you just got back from there 10 minutes ago, or that you most certainly did not steal their prized possession, or that you are not poisoning their food. My granny died, basically, of dementia, and before she died she would beat her adult kids who bought her groceries, because she thought they were poisoning her--again, a little old lady.
Please, please read Gavin de Becker's book The Gift Of Fear. He explains a lot in it how people, both men and women, try to be nice and kind, and it only gets them in trouble. I think the kindest thing you can do for this woman is call social services and let them know what happened. Think about it this way: If your crazy granny was going around threatening people and stalking children, would you want her possibly picking the wrong door to knock on and getting hurt by a very angry and frightened person? Would you blame someone who reacted badly to that threat and hurt her in self-defense? Or would you want her mental illness treated by nursing staff who are trained for exactly that situation, and her movements confined to a safe building? Either way, the granny you loved isn't there anymore.
