This doesn’t sound like a man who has fully committed to you. If he had to be told that he should be there for his child’s birth, and says one thing and does another — this doesn’t bode well for his commitment — not good. Not good boyfriend material, not good fiancé material, not a good potential husband.
I would take his attitude, disparaging remarks, and lying as indicators that he is not fully ready. I would highly encourage you to get some couple counseling immediately. His responsibilities to his child and to you need to be identified, clarified and changed. The question of weather or not marriage is in your future needs to be discussed directly with a professional. Find a couple counselor who can begin working with the two of you as soon as possible. There is too much at stake to just let this go — or hope that he changes on his own.