Wow. You've got a lot going on here.
I'd cut your mom out. Just because someone is capable of having kids does not mean that they'll be a good parent. She and your step father fall into that category, so, no, you don't have to see them or feel guilty about that decision.
Now on to your dad. You, unfortunately, were born to two individuals who struggle with substance abuse. You feel that your father abandoned you to your mother, and you're hurt by that. While I sympathize, what you don't seem to understand is that your dad has a problem. His brain wasn't functioning like a healthy person's would. That doesn't make what he did right, but as an adult, you need to start looking at your parents as individuals, not as your protectors. Does that make sense? He failed you, yes, but he failed because of a disease that renders him incapable of taking care of himself, much less a child.
You said that you hate being referred to as his daughter, but you also say that it hurts when you aren't included as part of his family. I think that you need to examine exactly what you want and expect out of him. You can't say that you don't want to be part of his family and then get hurt when you aren't.
Despite all of that, at the end of the day, you get to decide who is and who is not your family. If your biological family is too much for you to bear, then don't do it. If you're interested in mending a relationship with them, and I strongly suggest counseling for you whether you forge a relationship or not, then you and your dad need to work that out first, and then worry about everyone else (his GF, your DH, grandparents, etc.).