He is being a prize tosser. trust your feelings ...sleep deprived or not you deserve better! God I feel for you right now.. its hard as it is without your other half whining in the background about his feelings.
Do you have a good friend to offload on for some objective advice?
Yes the first few months are hard. My DS is 17 weeks. BUT.. that does not give him a reason to say such things to you.. To be honest.. when the baby haze lifts I doubt I could forgive him for the way he has been towards you and the things he says are just so hurtful.
I don't think he is depressed.. he is selfish.
Well done you for getting this far (HUG)