I do not know where to start. I was in an abusive relationship for five years so I speak from experience.
If my husband spoke to/about our child like that I would leave him immediately. It’s unforgivable. I can’t believe you think it’s your daughter who needs to apologise, she’s done nothing wrong and she was sticking up for her Mum. she’s probably confused as to why she has to apologise after the vile things he said about her!
If my husband blamed lack of libido of my being fat or repulsive I would leave him, how utterly disgusting, disrespectful and hurtful. And if he thinks that’s an acceptable way to speak about women do you want your daughter growing up around that?
If my husband messaged random women I would leave him - this is cheating, it’s emotional cheating. He did it behind your back and in addition she told nasty lies about you - for what purpose? Someone who loves and respects you wouldn’t do that.
I’m concerned that you are under his narcissistic spell, and you can’t see the truth because he has gaslit and manipulated you so much. The way he is treating you is abusive, controlling and manipulative. How would you feel if your daughter’s partner treated your daughter this way when she’s older? Your daughter is growing up seeing the way he treats you and learning that this is “normal” and that this is an acceptable way for a partner to treat her when she is older.
For what it’s worth, you are not the coercive, controlling one, he is. Nothing you have done is wrong.
Please, PLEASE get you and your daughter out of this. Children who are raised by narcissists usually have mental health issues, including many becoming narcissists themselves. If you stay in this relationship it’s will damage you and your daughter significantly. She will be mentally damaged and also will think that it’s ok to be treated the way your husbands treats you. You’re the only one who can protect her and she needs you to do the right thing. I left my abusive ex for me and my child’s safety and mental wellbeing and it’s the best decision I ever made. DM me if you like 🌷