Absent a custody, both of you have the right to custody of your child. The only way to prevent her is to get a custody order. The question for the court is what will be in the children's best interests assuming that both parents are fit? As the children are small, it may be hard for a judge to give you primary physical custody of the children. Daycare expenses are factored into the child support obligation. You do not indicate if the wife works or not. If she stays at home, why not let her watch the children during the day and eliminate this expense?
Concerning half the house payment, it will have to be decided what is done with it. You now have 2 households to support. Can you afford it on your own? Then buy out your wife's share of the equity and refinance in your name. If you cannot afford the housee without her help, then you both seriously need to consider selling the house. Whichever spouse makes the payments after separation should get credit for this in equitable distribution.
You and your wife need a separation agreement. You both need your own attorneys, even if you can reach an agreement on matters.
