Is my laptop fan ever supposed to turn off by itself and turn on again later? What should the laptop sit on?

Yes, the function of the fan is to balance the heat needed to the computer. It will turn on itself when CPU get white, and then turns off after sometime. That is totally normal.

As for surface to hold on your laptop, you can your your lap (lap-top duh), or table too, only pillow is not good idea, because you need to have at least 2 inches space element every the side of the laptop, and at least 5 inches space at the fan ventilation.

Hope this helps.