They may be talking about tax credits - cash lower income families get (I've never got them despite being a single mother, I work full time).
Long term you are better off going back to full time work even if you see no profit from it. Many men pay nothing even though legally obliged to so hoping your ex won't run away to Thailand if you split and instead will stay here and pay you is a bit of a risk.
Doesn'tmatter whose name the house is in or was bought by it is a marital asset - ditto savings. It is likely as you don';t work that if you can afford the mortgage you will stay in your home and your husband would have to mov in with his parents or find a floor to camp on unless it's a very big house and bigger than reasonable needs which I will assume not. However if even with some maintenance for you from him you could not afford the mortgage then the house may have to be sold and the proceeds split.
I remember when we had a baby, a 1 year old and both worked full time. It was very very tiring hardest phase of our lives. I genuinely don't think you should decide to split until the youngest if 5 and you can think straight even if you and your husband don't get on. However if you want to then make sure you know everything about your joint finances, pensions, seen his P60 each year etc etc so you have all the information.
As for how to get him out my children's fatherwas advised to stay in the home until the dirovce was final and until the finances court order/consent order was sealed buy the court and until he was paid his share of the proceeds, mortgage transfered to my lender will not youname and paymentmade to him. I suspect here your mortgage company will not let you take the mortgage over as you don't work) which will stop your husband buying his own place and putting up the children on his days.
You could pay for an hour's appointment at a solicitor now just to hear your rights - secretly from your husband, just to see where you stand. Dont' of course assume you get the children. No reason in 2016 why men cannot be the one children live with and that does happen ,