He is obligated to pay the child support until he receives an Order from the Court changing the terms of the original Order. If he has been trying to find work and has not been able to do so, he can make a motion to the Court to lower his payments.
If he just quit his job and is not trying to find work, he will have a hard time convincing the Judge to lower the payments.
Concerning your second question, although his name is not on the deed, he may still have an equity interest in the house if it was purchased with money earned during the marriage, even if you were the one who earned it. If it was purchased with money you had before the marriage, or received as an inheritance or gift, and kept the property in your name, he would not have an equity interest in the property. Nonetheless, until you receive an Order of divorce from the Court, he remains your husband, and you cannot simply throw him out on the street unless there is physical abuse that endangers you or the child. In such a case, you should get an order of protection and have him removed by the police.
Consider mediating your divorce. If you are in the NYC metro area, call me at 212-370-1660 to discuss.
Leonard M. Weiner, Esq. Divorce Solutions