You need to lose the mindset that he works and you don't.
He works outside the house for money, for around 40 hours per week, 5 days a week.
You work at home for love (of your children and, at one time, him), for many more hours per week, 7 days per week.
If you didn't look after your children, you would have to pay someone else to do it - a nursery, childminder, private nanny etc.
If you didn't do the housework, laundry, cooking etc, you would have to pay someone else to do all that.
It is therefore possible to put a monetary value on what you do.
You are also giving your children a fantastic start in life by staying home with them, and that is priceless.
It should not be the case that his family responsibilities end outside of his paid work. He should be doing his share of childcare and chores.
Moaning because the dishwasher hasn't been emptied?! He can fuck right off with that controlling crap.
If you do actually want to stay married to him, then list out all the things you do for the kids, in the home, for him.
List what he does in those areas, and for you.
List what free time he has away from the house, and what you have.
Then tell him how things need to change.
Oh, and condoms/vasectomy or no sex is non-negotiable.
Good luck.
PS: I had the same thought as a PP re loyalty. I'd choose the dog.
PPS: that comment he made about you doing it (childcare) all day, every day, so it's easy for you and so hard for him? Clearly, he needs to spend more time doing it, so he'll get better.