Am I, as the father, legally have the right to request a briefing of my case after it has been closed?

You may obtain your case documents from the clerk of the court where you case was filed. You will need the Cause Number from your case. If you don't have the Cause Number, you should still be able to find the case by searching for your name in the clerk's records. There will likely be a fee for the documents, but you certainly have a right to get copies of the documents. You may only need the final order, but if you need other documents, you should be able to get them.