This is SO very wrong. He needs to do his part. Somehow he needs to get straightened out and he should be doing his part of the household.
This has to be so stressful and overwhelming for you. And, my DH is sort of the same way in that his grandmother and mother used to do most of the stuff growing up. The only 'chores' he did was mow the lawn when he got older. Which left him pretty much knowing how to do nothing when he got older, went to college. He can't cook, has pretty much no cleaning skills. Thankfully I will get him started with things, and he has learned how to do some stuff over the years. All kiddos should help out...boys and girls and they should start young.
How about your older DC, I read you have a 7 y.o. Children can do some things as well. But, both you and DH need to model good behavior. They're going to grow up thinking this is normal.
Even worse, he is teaching your daughter how a husband should treat a wife. If it were me, that would bug me a lot.
You're supposed to be a team, and that means all of you work towards adding positively to the household. He doesn't sound like he's a team player.
Which is a great analogy that I use when I talk to my DH about this kind of stuff as he's really into sports! Some men are of the attitude that they "work" and their wives stay home and should do everything.
: Both of you should be putting 150% into your marriage and your family life, and it sounds like you're putting in your full energy and he's not lifting a finger.
And, something needs to *shock him* into reality, right now, to me it sounds like he's just letting the daily momentum carry him through his typical behavior.