You didn’t ask a question so I’m not sure what you hoped would come of your letter. I can only make some general comments.
I think you are probably right that your father has a significant mental health issue. What you describe is consistent with a personality disorder. However, it is also possible that there is an undiagnosed bi-polar disorder. I trust that when you talked to mental health providers, they’ve considered that. Whatever the label, the impact of his behavior on his family is tragic — both for him and for everyone else. He doesn’t seem to understand what he has thrown away. It feels terrible to everyone that they have all been rejected.
It may be that your sisters’ behavior is at least partly rooted in the fact that their father doesn’t love them enough to be a father and their mother can’t pull herself together enough to parent them either. When kids don’t feel valued by their parents, self-esteem suffers. They often then push the limits to see if someone cares enough about them to tell them to stop — and means it. So far, both parents are failing them in that respect.
As for your mom: She sounds totally overwhelmed! My hope for her is that she will get herself into some therapy. She needs and deserves the support. She could also use some practical advice about how to get back in charge as the parent of the household. You can’t be the family therapist and shouldn’t try. You will only frustrate yourself and alienate your sisters. What you can do is love them all and encourage your mom to get into treatment.
You could also research when and where support groups for single parents meet in your home city and pass the information on to your mother. I did a quick search of the web and found two that might be helpful to your mom. It’s often helpful for people who are dealing with challenging problems to feel less alone and to get support and advice from others who are going through the same thing.
You may also be concerned about whether a personality disorder is hereditary. As far as I know, it is uncertain how much is inherited and how much is learned. My guess is that you have strong motivation and support for not repeating your father’s self-centered behavior. In some strange way, a negative role model such as he is can be as powerful as a positive role model. You may well have sworn to yourself that you will never, ever treat people who love you as he has. That’s a powerful incentive to keep your own life balanced.
I hope this was helpful. I wish you well.