I have a studio apartment and the back wall where my headboard is I want to paint an accent color. The question is- my closet door is part of the wall, do I paint that the same color of the wall or keep it white like the trim?
You should paint the door with the same colour and design you paint on the wall. Otherwise the closed door will be highlighted.
You certainly can paint the door the same as the wall. The main problem with doing this is the door frame color. Do you keep it the same color as the rest of the trim or paint it the same as the wall? It would look better if the trim is kept the same as the rest of the room; base, door frames, casing, etc.
Traditionally all doors and trim are the same through out a room or house. But there is no reason to be traditional. Go ahead and paint the door the same as the wall and leave the frame alone. Now see if you like it. If not, it is easy to repaint.