I do agree with you that you should not have to pay them for something that they should be doing anyway. It makes matters even worse when you are not getting any support from your wife on the matter.
It sounds like your wife is extremely insecure. I have and ex that used to be that way too. Constantly accusing me of cheating on him. I worked from home and his mother worked for me so when he was at work I was with his mother. The last straw for me came when he accused me of cheating on him when I was out with his mother and grandmother. He knew for a fact that I was with his mother and grandmother too.
As far as her screaming at you like she does that is abuse. If you check the forums regarding relationships nearly all the posts are about women talking about how their spouse is abusive and you rarely hear about a man having to deal with abuse.
It sounds to me like you have tried everything that you can to make things better but you are not getting any co operation for anyone else in the family.
It is highly commendable that you want to stay for the sake of the kids and I do know how hard it can be. Is there any chance that you would still be able to have any kind of visitation with the kids if you did leave?
Is it possible to get into family therapy? If she refuses to go it is a fairly common practice that one spouse can start therapy on their own and then bring the family in to the sessions. Also if she refuses to go you can go for therapy for yourself and learn different ways to deal with her behaviour that will have to force her to act different.
If therapy is not an option I would recommed leaving. I do know that it will be hard to not be able to see the kids and I have heard plenty of guys say that they dated a woman with kids and fell in love with the kids.
One other thing, since you are the only one working, would your leaving have enough of an impact on her finances that she might agree to make some changes? For example if she has time to think about how the bills are gonna get paid, or how they are going to eat or she is faced with getting a job as opposed to doing some cleaning and having some respect for you.
Lastly it sounds like she is either suffering from severe depression or some kind of mental problem and that she seriously needs some professional help.