Your'e not being wet, its tough going isn't it. You sound quite isolated too. I would definately see your GP about how you are feeling too. You sound as though you need some quality time together as well (not easy I know).
In answer to your question, I think men/partners still need to do a bit in the home. They can sometimes have the easy option by being at work - avoiding the school runs, activities, shopping, general house stuff/cooking.
In a healthy relationship I think men should do something, not necessarily equal amounts of course, but something to help out. It shows they care about/respect you and the home you live in.
I think men underestimate being at home all day with kids, its the emotional thing too.
I'm dossing on here and mine is ironing! Just started doing it regularly about 3 weeks ago and its fantastic, I really appreciate it. He helps clear up after dinner and with the bath/bed routine too as well as unload the dishwasher in the morning.
I do the main share (tidying, hoover, cooking, paperwork etc) on a daily basis but we have a cleaner once a week to blitz the house.