What you say resonates with me as well as probably my daughter. I was raised in a similar household, although I was mentally, emotionally and physically abused, and I was the scapegoat child. My siblings were not abused physically, although I think all of use suffered from the mental and emotional abuse.
I always felt the same way you describe -- I can recall being suicidal at the age of 9, and wondering why I was alive!
I gave my daughter a different childhood from mine, and yet she may have the same feeling of not knowing why she exists at all. It must be something one is born with because she had a happy spiritually-oriented childhood, with an immense amount of love. It was just the two of us together, far away from family (thank God), and we did had a surrogate family.
I have a strong central core self, but she apparently does not -- she has a more ephemeral sense of self, and may be the reason she has to "prove" to herself she is here w/ cutting/starving/drinking.