Hi Michaela! ![Smile [:)]](/emoticons/emotion-1.gif)
My friend has just got his PhD and when he talked about the "defence" he simply said: 'to discuss a PhD dissertation'
Concerning the 'behind closed doors' issue: I find that it is good, but I've also heard the expression 'in closed session'
I have no clue about the 'essay' you are talking about, anyway 'report' sounds good to my non-native speaker!
I hope I helped you a little bit, anyway I advise you to wait for a native speaker!