Can you hang a ceiling fan from a bakelite electrical box or does it have to be a steel box. I am worried that the weight and vibration of the fan may pull the mounting screws out of the bakelite box.
A fan should be hung from a Fan Rated Box, that is rated for the weight of the fan.
Not all Fan boxes are metal, but I have never seen one that was bakelite.
You are correct that the box may not hold the weight of the fan, and it could come down on someones head, or cause a short circuit in the wiring from the vibration.
Electrical outlets sell an old work fan box. You could cut the existing box out and replace it with a fan box and do no dammage to the ceileing (that will not be covered by the new fan canopy.
Yes you must use a fan rated box that supports the fan independant of the box.