Also, the newly appointed custodial parent is in the military, can they be barred from taking the children out of the country or more than however many miles from the other parent?
Estate Planning Attorney
An appeal is not filed in the same court which issued the ruling. It is field in the appropriate appellate court; in Georgia that would be the Court of Appeals or the Supreme Court. Which court is determined by the courts' jurisdictional laws and rules and the type of case that you are appealing.
To appeal, you must generally show the trial court committed error. If there is any evidence to support the ruling, it will usually be upheld.
Your question sounds more like you are asking for a recusal of the trial judge. You must have evidence of some sort of bias against you, not just that you don't like the ruling made.
As to your last question, a court in a Georgia divorce or custody trial can include reasonable restrictions in a child custody order. It cannot however, prevent a parent from moving. Your remedy in that event is more likely a modification action.
For more info on Georgia custody appeals, see http://www.gafamilylawblog.com/appeals/