Dear K.,
My ex-husband was the exact same kind of person - very judgmental and critical of everything I said and did and my appearance was never good enough. People like this actually have very low self asteem and the only way they can feel big is by making everyone else around them feel small. He told me I could never survive without him and since I divorced him I got my PhD, a very good job, a wonderful loving husband and a beautiful 7 week old baby girl.
It is very demoralizing to have someone like that in your life and neither you nor your daughter need that at all. Thirteen is a very bad age to have someone tell you you're not good enough, you need support and reassurance in your teen years more than ever. You can always try to tell him that this behavior towards your daughter is unacceptable. If he's anything like my ex, he will not toleerate being told what to do. If that's the case, then you need to cut ties with him completely. It is better for your daughter to grow up without him than to have such a negative, selfish, demeaning person in her life. Chances are, you will feel better about yourself as well without him in your life. I too told myself that the Christian thing to do was to forgive him and try to stick it out (for 6 1/2 years I tried to forgive and forget). But trust me, God wants you and your daughter to be happy and mentally healthy. You have done more than enough to try to have him be part of your daughter's life, but it seems like he just doesn't really want to, at least not for the right reasons. If he loved and cared for your daughter, he would not behave this way.
I would speak to your daughter very candidly about him and let her know everything (if she doesn't already). You should not try to protect her from the truth, she is old enough to understand. Have your daughter mail him a birthday card and not speak with him directly if she still wants to wish him a Happy Birthday. That way she can say what she feels in the card without him lashing out at her over the phone (if that's what he does).
Good luck and please take care of yourself and your daughteer, not him!
N.