E.S.
I used to work as a paralegal for mostly family and real estate law. Here's a few pointers on "how it generally works"...
1) Yes, don't ever leave the house. You won't get back in.
2) The idea of keeping the same house for the sake of the kid's stability is a good one that the judges will follow but only if it makes feasible economic sense to do so.
If your husband makes a whole lot of money and is able to maintain two households without a problem, it may happen. I generally only saw this happening where the husband made way in the upper six figures and the kids were older (already established in school)
If your husband doesn't make enough money to really support two households, it probably won't happen. Generally, the house is sold with the proceeds split to start over.
Getting lifetime maintenance (formerly alimony) (which is pretty much what him paying for the house is) is near impossible, again, unless your husband makes a ton of money and you were married forever and don't have any skills. Younger people who just didn't work are pretty much expected to find meaningful employment and then daycare costs come into play unfortunately.
My honest advice is to stay in the house as long as possible, but don't "fight" for a house that you can't afford yourself if your husband really doesn't make enough money to support two households. You'll end up wasting more money on lawyers than you can imagine. Be realistic in your expectations of what your husband can and can not provide. In other words, he isn't expected to move into a $400/month studio apartment just so he can pay for the house forever. He is supposed to be able to afford the same type of house (think that your children will have to stay in his residence also) as well as your old one.
In reading this back it sounds a little negative and I meant this to be helpful and not meant to be harsh or insulting in any way... sorry if it comes across like that and now I've got a jillion things to do and can't figure out how to quickly fix it. These are just generalizations...if you want to pm me, I might be able to help more with knowing more about your actual situation. I worked for a bunch of divorce attorneys, from the regular suburban ones to the high powered loop ones and have pretty much seen it all!