Can I have a court appointed attorney and a hired one at the same time for two different cases?

Please understand a person gets a court appointed attorney when one can lose their liberty (criminal case), but does not have the right to one when it is just a right they wish to pursue (custody, support or such.) So the answer is she has two attorneys for different matters for different reasons. If she cannot afford the attorney to pursue the matter at dispute between the two of you, she can pay the attorney and he can ask the court to order you to pay her back if the court determines you have the ability to pay and she has the need (like you are being difficult and needs the advice and representation. So be careful if you have the money to get ordered to pay that your position is a correct one based in law and the circumstances of the case.

Yes. You can be poor and qualify for court appointed representation and still decide to use your limited funds to hire a private attorney.

Yes. One has nothing to do with the other and lawyers are appointed only in criminal cases not civil cases.

Yes, that commonly happens. You can only have an appointed attorney on a very limited number of kinds of cases (of which child custody is not one).