Please do not hate yourself.
I don't know why you do. But you have been incredibly strong to keep yourself together through all of it and still making plans for your life. You should be so so proud of yourself, to have gotten this far, and to not seek any kind of payback. You are better than your abusers, and you shouldn't forget that.
I was abused by family members from the time I was 6/7 till I was 12, at which time I left the country, and escaped from it all. I am from a country where family is "the most important thing" too. Younger people are not allowed to talk back. I even wrote about my abuse in a diary, which was opened and read after I'd left the country. Funny thing is, my grandmother was the one who read it, and didn't believe what I wrote. WTF, why would I lie in my own diary. But anyway, that is not the issue.
I don't often go back to visit my home country, but when I do, I have to face my abusers. I do my best to ignore them at family gatherings, and make my appearance brief. I guess, for you that is more difficult, as they are your immediate family members.
I don't think I have ever quite dealt with the abuse fully. I have just learned to accept it as part of my life. When I was 16, I was very lucky to meet someone to whom I could speak about my childhood in every detail. That person helped me so much, in understanding that none of it was my fault, nor ever will be. He told me I had incredible strength to have held it in for so many years, and to have coped so well. So now I would like to tell you the same. It is inevitable that you will feel sad about it for the years to come, but it doesn't have to make you feel scared, or doubtful about who you are.
I thing the most important thing for you right now is to focus on your studies. Once you get in, there are facilities at uni where you can speak to counsellors about your problems, and they will help you deal with your past abuse. For now, you just have to know that you can do it. When you graduate, you will have your own life, completely separate and independent from them. You can always go back home to see your siblings and cousins, but instead of staying in, go out so you can be away from your parents!
I am glad I got away from it all, and that the experience has taught me to be a stronger person and cope with many other nasty things in life. Right now, my life is exactly how I want it to be, and there is no reason why you couldn't have the life you want. If you would like to have a chat privately about my personal experience, quote me and I will reply!