Gosh he sounds hard work. I think you're handling it very well. His children and the house and the responsibility for the dog can be shared, and it sounds like you are doing everything it is possible for you to do. He can moan and criticise, or he can take the dog, or he can hire a dog walker, or he can do something else that suits him. It sounds like you have a lot on your plate and you are doing your best. It isn't realistic to expect a family life to carry on totally as normal when one of the adults is not there.
I don't like the idea of him checking up on you - asking how often the dog had been walked. It feels like a very parent-child interaction, him wanting to know if you'd done your chores. But perhaps it wasn't like that, and I'm just reading that into it as I know that dynamic so well from my own interactions. In any case, I think you handled it assertively.
