J.,
You have got a tough problem there. I hope things work out well for you and your daughter.
My only suggestion is in relation to the public toilet issue. My three who are potty trained dealt with the same issue. The only solution I found is to take one with you.
There are portable toilet covers - some of them even fold up and come with a carrying case - and I bought mine on eBay for about $10. Search the web. They are everywhere. We used the fold up kind with my daughter and the soft kind that removes from the baby potty with my boys. Keep it in the car and take it with you in a bag.
You will have to acclimate her at home first. Give her the time she needs to get used to it. Then explain that she can take it with her wherever she goes and it is for kids. (Smaller hole = no falling in.)
The first time you try and tackle the public toilet, I would leave your son with someone so your concentration can be on her. Go someplace where there is not a lot of people and take however long it takes for her to get comfortable and go.
I made the mistake with my first child that when she screamed on the public toilet, I immediately took her off and took her home to pee. What message did I send? The public toilet is big and scary. Mommy had to save you from it and whenever you don't like the toilet, scream as loud as you can and mommy will get you out of there. Not a good message.
So I came with my positive reinforcement (cell phone to call daddy and some candy,) made sure I was somewhere there wouldn't be a line of angry people waiting to use the toilet, and went. Yes, she threw a very long tantrum. Probably 30 minutes the first time. But eventually she peed. The second and third, maybe the fourth time, she still threw a tantrum, but each time she learned that the toilet wasn't going to swallow her and that tantrums didn't work. Eventually she went without a fight, then eventually without the seat.
After that, I knew to do things differently with my boys. Once they were potty trained at home, I made a special trip with them some place alone (potty seat in bag) and we used the potty. There was a little resistance, but from past experience I learned not to encourage tantrums. Things went much smoother.
All of my children have also been terrified of the self flushing toilets. You can put your thumb over the sensor or if you need your hands free, bring a roll of masking tape and cover it until you are finished. We also referred to these as "the magic potty" and my children eventually learned to see it as such. Now it is a joke with my big kids.
For your other issues, you may want to take your daughter to see a professional. She may have some sort of sensory issues - that would explain the vaginal annoyance and the lack of desire to poop.
Best of luck,
S.