The jury box 'balustrade' - just a balustrade, or does it have a special name?
Usually just jury box. We're not smart enough here to call it a balustrade.
If the judge asks for something to be fetched from another room within the courthouse, who does he ask, a clerk?
He'll ask either his clerk or the bailiff.
The main aisle than runs down the centre of the court from the main door to the bench - just an aisle, or a special name? (I imagine it's just an aisle, but you never know.)
Just an aisle.
In NY, how is the evidence catalogued? I currently have it numbered. Is that right or should I use letters?
Doesn't matter. Whatever you can get cheaper at Officemax.
If one of the lawyers accuses the other of playing to the court, is the term 'showboating', or 'grandstanding'? Or something else totally?
No, there is no real objection to "showboating" unless there is an actual breach of the rules of evidence. What the attorney can do is call for a sidebar (a meeting with the judge and the hsowboating attorney that is not heard by the jury) or ask to approach the bench (same thing as a sidebar really) and tell him that the other attorney's activities are in violation of the ethics rules. The judge may give a warning but probably not anything because there is no legal objection.
Again, is this the correct terminology - '... move for a continuance...'? (The phrase a lawyer would use if he suddenly realised he could be better prepared in some way so wanted to postpone the trial.)
You would move to adjourn and give the judge your reasons. Requests to a Court are done through "Motions". You could move for a continuance, I suppose or ask for one, if there is a real reason, but you ought to look up the rules of procedure for New York. The New York State Bar ought to have a link on its sight or can point you in the right direction with just a phone call.
The person who records a transcript of the proceedings - the court stenographer?
Court reporter.
If it's 'your Honor', is it also the Defense, the Prosecution (or the People), etc? Are they all proper nouns, so capitalized? [But surely it isn't the Judge?]
It depends. When the judge speaks in a formal speech, often times he will simply say the Defense, the Prosecution. When he doesn't know the attorney's name, he will say the Defense or the Prosecution. If he's a golfing buddy or well known, he'll probably use the attorney's name unless it's a formal hearing in front of a jury. Then it's more likely he will use the Defense or the Prosecution. When the attorneys refer to themselves the refer to themselves as the Defense or the Prosecution. When they refer to the other, it's usually the same. SOmetimes they just say "dickhead", but that's a matter of personal choice. These aren't hard and fast rules, more like general practice things.
Yes, capitalize. You're using it as a proper noun.