I think she defo has something psychological going. The autism world explain a lot. My DD has adhd and self harmed because she was so frustrated she didn't know how to fit in or be like other kids and can't keep up. So girls mask it and the most effective mask/defence mechanism is hate and rudeness/backchat.
Re her going out in the rain, you can try to bring her back but sometimes when my daughter is angry she will do worse and worse things if I am there to see it. She wants a big reaction. If she's sad she'll allow me to help but if it's anger she's doing it to provoke something. If I walk away and appear not to have noticed she quickly figures out there is no point and calms down in her own and comes back.
This thread will give suggestions but you can't take anyone's advice because it's not the what she is doing that matters, it's why she is doing it. Without that you're stabbing in the dark.
Don't let her abuse you verbally or physically though. Not just to protect you but to protect her. The chronic shame that follows for them if they have hurt you makes it worse and makes it a cycle. There are different ways and means of doing this. One is to say no I wont have that, and walk away. Another is called something like non violent confrontation - Google it, it's a way to stop the behaviour without it escalating or leafing her to entrench. Any child psych will want to understand what the pay off for the behaviour is, if it's attention and a drama and you give that then perversely you are rewarding the behaviour.
But if she has autism, or is a victim of something external the response might be different. You know her best.
Finally it is ok to take a break. Leave her with someone or make some kind of arrangement to tie her up away from you so you can take a moment to rejuvenate.
They say even the most successful SAS soldiers only survive if they make time for R&R. Anc this isn't much different, it's a war of a different kind but the mental techniques are the same.
I'm with you. I'm going though it too. It's awful and hard. And you get pushed to your limits. She sounds like she wants to see if she can break you, but you mustn't let her do that because she doesn't understand what she's doing and she needs you. Xxxx