Like the owner's manual says it does.

Or are they supposed to? Not a huge deal, but it would be convenient... It turns on and off in the conventional manner. Before I start investing any time in this, are 280Z dome lights intended to stay on for a bit once the car is shut off/door opened etc.?

My dome light turns off as soon as the door is shut, it is not on a delay at all. If you push the button at the door sill that controls the light, either driver or pass. side, its on/off and thats it.