You need to speak with an attorney in New Jersey. The criminal case is a problem for you, if you are trying for a government job, domestic violence type cases are a real problem for a government to ignore. If she is not going to testify at the criminal case, that might not stop a conviction based on police testimony. She could even be charged with obstruction of justice for not cooperating. You need your own attorney so your attorney and hers could get the prosecutors to drop the criminal case against you. Then the next step is a formal dismissal of the restrainign order by both parties. After that, then bring all your records to any interview to show there are no pending actions against you. As to what shows in a background check, any filed case is findable for a background check. Don't ignore it, act positively for your future.
This is a general answer and does not address the specifics of your individual case. To give the specific answer you need our firm needs you to come in for a consultation.