These are not minor complaints. He's not being a father to his child, an equal parent. You work full time and do 99% of the work (not exact math, obviously) taking care of your child. And his complaint is that he shouldn't have to do chores (all of which YOU do) while working full time? (Which you do TIMES TWO AT LEAST because you're also taking care of the baby!)

When the f are you supposed to get anything done for yourself?!

Please tell us he doesn't also go out with his friends or go golfing.

This is who he is. He has shown you who he is. Believe him. It sounds like you'd be better off without him; after all, household chores would actually decrease for you without him around to make messes.

Please remember that the example you guys set for your kid will make a significant impact on who they become as people. If you continue to put up with this behavior, this is what your child will deem as being acceptable -- whether to be treated like this by their partner as he treats you, or by treating their own partner like he treats you.

If you've already tried talking with him, tried counseling (or tried to ask him to go but he's refused), tried showing him everything you do in a day while he does jack shit once he clocks out of work while you run yourself into the ground caring for him and your child, you've got the facts. You know what the deal is. Is this what you want indefinitely?