Oooooook. I KNOW this sounds like a ridiculous question. Is all of this normal, do other people fight like this too? How often would YOU tolerate full blown being yelled and cussed at? What should I DO!
If you have read this far thank you a MILLION times over. Help!
Whether or not this is normal, it is definitely unhealthy for you. What he is doing is called emotional abuse.
It sounds to me like he's got an anger problem, not just with you, but with his child as well. Would he consider anger management classes? The more you tolerate, the more this will continue. It will escalate, and will probably develop into physical abuse (as trends generally indicate). If you have a chance to stop it now, you should.
Many of us on this board have been through similar experiences with partners who take anger out on us and then try to excuse it or justify it by saying we need to toughen up or by pointing out our flaws to divert attention from their bad behavior. It isn't your fault and it isn't normal. He also seems to be saying "you're not listening" when he means "you're not obeying", which is troubling. He may or may not understand what he is doing, but it is negatively impacting you and I can bet it's harming his daughter, too.
Are you in individual counseling? Have you done any research on emotional abuse or power cycles? I would suggest doing so (in private browsing mode, if you both share a computer; you want to avoid that volatile temper of his). Please see a counselor because his behavior is definitely abusive and you need to talk to an objective professional about this to learn some coping strategies, if not get advice on how to remove yourself from this situation. Please read the book "Why Does He Do That?: Inside the Minds of Angry, Controlling Men" by Lundy Bancroft.
I think it will help you get some perspective on your situation. It's easy to read and informative. Also consider reading "Should I Stay or Should I Go?" by Lundy Bancroft and Jac Patrissi.
You're posting here, which means you recognize that something isn't right. Pay attention to that feeling. Try to figure out what you think is acceptable and what isn't and try to figure out if it is possible for you to have a relationship with this man that isn't guided by intimidation, fear, and domination. You don't sound happy. You don't sound scared, but if this escalates, your fear will, too.