Amfleet,
I can't answer your question of what happens at night in the NOL station, I didn't pay any attention to that while I was there. I did pick up a few local guidebooks though, so I'll take a look through them just to see if they say anything about hours.
However, might I suggest that you change your plan and instead do what I did? I would suggest that you fly down with your family, then leave early so you can take the train home. This would permit you to take the Sunset to NOL, then take the City of NOL to Chi, and from there take the LSL home to Boston.
Setting your trip this way would almost guarantee that you wouldn't get stuck in NOL late at night. The Sunset is due to arrive at 9:20 AM in New Orleans; the City of NOL leaves at 1:55 PM, giving you a layover of 4 plus hours. Short of a major problem, the Sunset is usually not that late getting into NOL from Florida. Even then they have a 3-hour layover window at NOL for the Sunset. There is also a huge amount of padding between the Bay St Louis stop and New Orleans. Note that the Florida bound Sunset only needs an hour and 25 minutes to cover that distance, while the NOL bound Sunset gets almost 3 hours for the same distance.
So your odds of making this connection are very good. Additionally the connections in Chicago also allow plenty of time.
Now I did leave the station while I was in NOL, however I only went towards the St Charles Streetcar Line. I didn't wander in other directions. I would echo what others have said, during the day it's ok. However night could be a whole other matter. I also did not see any hotels in the immediate area.
One also has to be quite careful even walking in this area, as there are very complicated intersections and highway off ramps. So unlike my typical New Yorker cross against the light mentality, even I was waiting for some of the traffic lights in this area.
The first class lounge is called the Magnolia room; it has one TV, a small coffee pot, and a bottled water dispenser. You can get food at the station, although the choices are limited.
If you have any more questions just let me know.