I'm sorry to hear about your father and his violent behaviors. Yes, it is possible to develop PTSD (post traumatic stress disorder) after living with someone who has alcohol abuse and violent behaviors. It sounds like you witnessed domestic abuse which you may have felt was life-threatening. If you are having nightmares about these traumatic events, then you may have PTSD.
Please follow-up with your doctor or a mental health professional to get an assessment into whether you meet criteria for PTSD.
In the meantime, you can see how severe your anxiety problem is by clicking here.