A man is paying the morgage for the home he and his ex lives in, would he also have to pay child support for his child who also lives there with him?

Only a portion of the money you pay in mortgage is for the child's shelter allowance, so yes, it is completely proper for a judge to order the father to pay child support in addition.   Although you can always appeal a judge's order, if the judge spells out in the decision and order the basis upon which he/she granted the mother support, and the decision is rational, based on the guidelines and what you're earning, I'm not sure there is anything to be gained from appealing the order.