@SheffieldCrown - I don't have lots of time right now, but I feel for you. I know from personal experience just how hard this is. Please don't take anything what I'm about to say as criticism.
There's a few things in what you've said. Firstly, you need to be paying attention if these little incidents are prone to happening. No being distracted or not watching. It's not fair to your youngest DD. It's also entirely possible that your older DD didn't mean to hurt her and is mortified at the tears, as well as it being a sensory trigger (crying is loud and unpredictable). Watch and prepare to distract in a casual, low key way.
You need to learn ways to take the heat out of the situation. With PDA you can't be confrontational in any way or else things will ramp up in the most horrible way.
Walking away from an escalating situation is an acceptable solution.
Shouting never helps. When they get louder, you get quieter. You prepare yourself and keep reminding yourself : quiet voice. It's not just about modelling the right behaviour, although there's an element of that, it's also about adding stress and anxiety to a situation which is what a loud voice will do.
Also, I absolutely wouldn't be driving her to your parents house on her demand. Give her time and space to cool off, don't crowd her. But at the same time allowing her to demand unreasonable responses is not OK. Stay calm and don't engage. She's there now so there's no point in escalating things. But I would send her a loving text to say that you miss her and you are looking forward to seeing her again.
She needs lots and lots of love and understanding. And I think you desperately need some more info and support around autism and PDA.
And just to emphasise, absolutely no judgement here and no criticism implied. It's really really hard work and exhausting parenting an autistic/PDA child.
If you put them on a spot with a question like "what did you do?" "what's happened?" etc, it's going to be really hard to get an answer and you're likely to be met with avoidant behaviour like anger. Try to avoid asking these kinds of direct questions, especially in the heat of the moment.
Re things like being late for school - her reaction is completely understandable. The attention and shame of being late would make that overwhelming.
My DD lies. All the time. And about the weirdest, strangest things that don't give her any advantage/benefit. I know they're lies. I think for her it's about making her feel in control which reduces her anxiety. I don't call her out on them - there's no point. She gets flustered/angry/humiliated if I do. I really do think it's about reducing anxiety. And that's what a lot of autistic/PDA behaviours are.
I'm a big advocate of "all behaviour is communication" school of thought. She's not being naughty or bad, she's just in desperate need of help.
Re the tapping - does she like fiddle toys? It really is about pre-empting behaviour and being ready to distract and divert in a casual way.
I'm sorry, I know this is really hard work. It's hard keeping all the balls in the air all of the time. I think if you understood her more, you'd feel more empathy with what she's going through. She doesn't want to be like this. She needs lots of love, praise and more than anything she needs your help. If she sees the dislike/detachment mirrored in your eyes, she's going to hate herself even more.
Just reading through your post again you seem paralysed by indecision and scared of upsetting them. Are there any autism-specific parenting courses you could go on? We were offered three and they really provided so much insight.
Don't engage in battles. You make a decision and stay calm. You don't try to stop her chasing her sister, you stop her. Not by screaming. You show her empathy when she's screaming. When she's yelling she hates you, it's a calm voice you need, tell her you love her and you're sorry she's having a rough time right now. That you understand she's feeling really angry so you're going to go in the other room and give her some space, but you're there if she needs you.
Preventing incidents really will be key though. You'll need eyes in the back of your head and a list of preplanned diversion tactics.
I speak as a mum who's been through all of this with two DC and it works. Most of the time. And it takes a while.
Also you need to take care of yourself too. You can't be driving around so late you're so exhausted you can't get up for the school run.
And I'm sorry your ex is a shitbag. That's the last thing you need.
Sorry, last thing! The right school setting will make such an enormous difference. It was like night and day with my DS. The threat of school just looming all the time eats away at their resilience, leaving them right on the brink for everything else. Even at the right school it can be extremely exhausting for an autistic child. Although perversely, sometimes they need the structure and struggle more during the holidays!
It's going to take a lot of hard work OP. And you're right, there's sod all support out there. You could try social services - tell them you're struggling and they might be able to help. But that's a route not everyone wants to go down, so up to you.
Good luck and please keep posting. You're not on your own x