"Have you tried just taking a deep breath and thinking 'actually, I don't care'? Just block his huffing out? That is hard but it works for me in all sorts of situations - like just relax and admit 'I can't fix this' and smile. He might be astonished that he's not got you on the ropes anymore and the new dynamic could feed into some talking?"
I think we are on the same track here. 
I don't think kicking off is a good idea just yet (as you don't know what is likely to happen), but that may work in the future.
I used to go on strike sometimes and refuse to do the ironning. Difficult, life without an ironed shirt.
Not caring is the key. He gets at you because he is like this, but he can't get you if you don't care. So for your own sake, just don't care. If it is good enough for you, it is good enough for him and otherwise he should do something about it (which he does). It may not be as bad as it seems...
So, as you have now offered a solution for the dishwasher, be consequent, but no longer offer any solutions any more for further issues. Just carry on as normal and if he doesn't like it, tough shit. (sorry 
). I think you did well in that e-mail. He hasn't replied, but that doesn't mean he hasn't read it. Let it sit for a while. If he says or does anything bad again, leave it and tell him later that you really did not like it and don't mention it any further (this is not 'I don't understand why you snap at me', but in the event that he were to shout at you in front of the children for example). Don't be angry, don't be emotional, don't kick off, just mention it. That worked with my husband as he started to slowly see that such things played on my mind although they didn't on his. He claims he has forgotten and blocked it out. I sometimes wonder but it is possibility.
In particular things like shouting close to your ear (has happened to me too, back against the wall and shouting in my face) is frightening. But don't tell him that, tell him you could actually have become deaf because of this (which is true). That is if it is still relevant. That it is not acceptable and there is an end to it.
I can imagine that you don't want to do this for the rest of your lifetime, I wasn't considering that either.
Is there any chance of your daughter and son doing something really cute, or doesn't he take to that? In that event, stay well away and see how he does, but make sure your daughter doesn't become too clingy. And stay around so you can go and 'help' him if he needs you (for the emotional stuff).
I think the weekends are maybe a good time to 'include' him (if he is tired during the week). It is possible he feels that you are excluding him because the children (particularly your daughter) will come to you (as they see you more often). Next time your daughter comes to show you something or comes to tell you something, ask her to go and show daddy too, because daddy will also be interested. Make something for daddy? That might melt him (not visibly 
). If this works, you can try the 'tonight, daddy is really tired, but tomorrow daddy will read you a story, won't you, daddy?' I think this may make the wrong impression at this point. Don't ask him to come and help, let the children request his presence. If it doesn't work, or if it is liable to end in tears from their side then don't. If he does clean something up or put one of the children to bed thank him for it, maybe ask him how it went.
Maybe you could plan an afternoon or a few hours out. Let's say go for a walk. Ask him if he would prefer you to leave the children with him or to leave only one, or take the two of them with you. Maybe he would like to come? Don't know if this is possible, but as he doesn't want you housebound and wants quality time, he may just have enough self-confidence to make it through the afternoon. (if it is a self-confidence issue)
You could, down the road, try to buy him a book on stress management and leave it in a strategic place or a leaflet. Only if you feel confident he is wondering about himself, though.
I feel sorry for you and you should know what you want to do: walk or stay and change. I stayed because I felt that the man I had met was gone and that he was somewhere in there. I doubted at some point that it was going to work, but we moved to another country, to a job with more money and fewer hours and hey presto! Gone was the stress and back was the talking. I don't know whether that is applicabe to you, but things can change.