Your mom didn't make a mistake. She knew exactly what she was doing. Stop romanticizing and sugar-coating the circumstances because it steals your power.

Your dad and the responsibility for the situation he created is on him. I had a situation like this unfortunately, so I can speak from experience. You can't blame his wife for how she is handling things. It isn't fair, but it is reasonable and you'll understand the betrayal once you have your own family. Their family is trying to cobble together some sense of normalcy and you being there keeps that image from ever being achieved. Your father not only cheated on his wife, but his whole family. Imagine her pain. Finding about my father's infidelity was one of the worst days of my life and meeting her was pretty bad too. It has taken years to get to a place of mature detachment and recognizing that it wasn't her fault. Don't expect presents, open arms, or to share in family events; that was your mom's choice when she had you and the wife shouldn't be expected to be superhuman. She shouldn't be directly in contact with you. It is really poor form that she sent you that letter. You didn't deserve that. And, you shouldn't ever be in direct contact with her. I will never bring my sister around my mom out of respect. And, I still have to fight the urge to strangle her mother every time I see her. She's one of the few people I hate and always will. That said, family is a choice and you were fine before and you will continue to be fine. You are not defined by their choices. The best revenge is to be awesome.

Tl:dr: The adults need to step up and give you guidance. Slowly develop a relationship with your siblings if you want to and if it is good for you. This is harsh, but life is not a fairytale. You are not entitled and don't expect anything more. That said, be awesome in life.