If it's any help, my situation is very similar. I've had 4 week sulks from my husband, ending when something happens to break his mood (often someone else making him even more cross than he was with me, but sometimes a nice thing). He is always critical of me, and usually also cross and nasty, mainly over the state of the house. He can't cope with untidiness, but tidiness is very low on my list of priorities.
Our children (aged under 6) are aware of the problems (on very bad occasions, their dad has addressed his complaints about me to them, telling them "Your mother's done ..." or "Your mother's a ...") They argue with him in my defence, and often ask him not to get cross. They talk to me about it, and I tell them it makes me sad, and that it is wrong for him to get cross and rude. I do say that we love daddy anyway and that he loves them very much too, which he does.
We tried a couples' reconciliation programme, which went well until near the end, when something that I said made him go in a mood, and we never put any of the ideas from the programme into practice. He won't go to a counsellor with me or without me.
So, as everyone here is posting to you, in effect there is no relationship in the normal sense (not sharing a bed, and doing nothing together unless it also involves the children).
But I have decided not to end the "relationship" for now, because 1. I don't want to be without the children every other week 2. I can't believe that for our children to divide their time in half between 2 homes and 2 separate parents would be good for them, 3. I am sure that it would be hard for them to spend every other week alone with their dad and his mood swings and tidiness obsessions (and his rudeness about me), 4. I wouldn't want to go through a nasty legal process invoking all his faults (and he mine) in a court, and 5, as you write, there is always a danger that he might bizarrely end up being given full time care of the children.
So my idea is to stay, spend a week or so away every couple of months, try to surround our children with friends and relatives who have more normal relationships. Possibly mad, but am trying that.
Good luck with your own attempts to resolve things for the best.