Do Hague Convention proceedings decide which parent gets custody of the child?

No .The principle is that custody will be decided in the 'home' country after the child is returned. Courts hearing Hague cases decide whether the child should be returned to the country where he/she was living prior to the abduction. If, however, a return order is granted, the court can decide that if the removing parent refuses to return with the child, the other parent is entitled to travel back with him/her instead of that parent.
Convention In Action