Horrible behavior towards loved ones

I am a young female (16). My mother left me when I was about 9. I can still see her, she lives in the same town, but I don't live with her. I live with my younger sister, my dad and his mother, my grandma. They don't torture me, they buy me stuff like sweets, clothes, also sometimes cosmetics or when I don't have enough money especially grandma lends me some. They treat it like a debt, but I respect it in general. But my dad drinks a lot of alcohol, and all three aren't ashamed to insult me. I've heard from them my life sucks, I'm nothing or that I won't finish the school. My father often calls me a moron, and other vulgar things. He screams at me for everything, even though it's nothing that horrible. He also hits me, mostly hits my head with his fists. I think I deserve it, but he's very strong, so it hurts very much, and I sometimes have a bruise. Today he slapped me with his shoe and nearly hit my eye. He was doing it even when I was little. He just grabbed my arm and shook me like if I was a rag. When I was really small, he hit my mother because we wanted to visit together my friend who celebrated her birthday. Also one day I was watching TV and I asked him if he knew a music band that was on here. He screamed at me that I should've stopped saying that ****. So we don't talk normally to each other most of the time. He also controls me, but I don't need it anymore. I'm not a little kid. I know what I should do.
I also probably suffer from several mental issues like anxiety, paranoia, depression, maladaptive daydreaming, panic attacks, OCD and PTSD. I don't do self-diagnose, I just suppose it. But even if I didn't have any serious problem, I just don't feel good. I was on the bottom in summer and it lasted about a month. I was crying a lot, I couldn't eat and I wanted to stay in bed all the time. My grandma didn't understand it. She laughed when I cried and she insulted me. She couldn't get that even a 15 years old girl can feel mentally horrible. I talked to my mother if she could find me a therapist. She laughed at first, but then agreed. Later she thought I'm already okay, but I wasn't really. I was just doing something that distracted me and I liked it. So then she said we would go to a professional. I was really excited about the thought I'd be good again. But my father needed to know. So I told him. Both, him and grandma, started to scream and they threatened me with a thing that actually caused me that I'm so worried about my privacy and I always get scared when someone comes to my room. So it was horrible. But I made it, because the school started so it distracted me.
They two aren't supportive at all. I like music, so I decided I could start playing an instrument. My favourite is a bass, so I told them I could start playing it. They laughed that I won't make it. I had a friend on the Internet. He's never seen me but says I could make it. So it really hurts me to hear from my closest family I can't do something, when people who don't even know me, supports me.
My biggest struggles are with handling my emotions. I'm just too impulsive. I do something before I even realize what I'm actually doing. I have my history of self-harming. I have some scars on my hands from sharp things. I also hit my head against the wall when I'm really triggered. Or I just go out and I smoke to call myself down. It may lead to my drug addiction in the future life, but I think it won't happen. But what parents do to me and make do isn't the subject of the questions.
It's what I do. I'm very agressive towards people at home. I sometimes hit them. I don't hit them with anything like chairs or threat them with a knife. I just use my hands, and because I don't do any sport and I'm quite small, I'm pretty weak. Or I just push them. It's like only once and then I stop unless I defend myself from my father. I don't wanna hurt them, I really love them even though they do what they do, but I just react so quick before I even realize it. I would never cut them or really hurt them. I'm also vulgar towards them. I don't just go and hit them, it's always when they hit me or when they turn off me the Internet and things like that. I mean, the Internet or some drawing is something that keeps me distracted, so I don't feel the bad feelings. And I just like it, I must confess. I don't listen to loud music, I have always my headphones on and I just play something. For my bad behavior I need an impulse. When I calm down, I realize what I did and how bad it actually is. I hate myself for it. I'm trying to control myself more now, but sometimes I don't handle it. I think I first hit my father when I was ten, but he was doing almost the same stuff, I think. I don't remember it, but I knew I realizing I was doing horrible stuff even back then. I love my family, I want my grandma to go to the doctor, because I suspect she has skin cancer but she refuses and says I'm mentally ill. But it doesn't matter. I'm afraid every day that they'll die. But sometimes I just don't handle the rage. I'm a very horrible person and I hate myself for this. I wanna stop it, because it's bad and it could affect my future life. Actually, some of my interests might possibly make me very successful, but if this goes public, it could destroy me and my life. This is haunting me and causes me panic attacks. Later in life, I'd love to be a good person and help people, so now I'm trying my best to be good, but I've done this bad stuff already, so I can't change it.
Can it really affect me later and destroy me? Can people start hating me for it if they knew what caused me this behaviour? If not, how can I stop being worried about this? Should I be hated for things I did as a kid, and as an adult I'm good?
I'm sorry that it's so long, but I needed to make it clear. Thank you for your answers.