In which case the heated part will cool faster if you put it on a wooden stand or a steel plate. why?

The part will cool down if you put it on a steel plate, because the thermal conductivity (the ability to conduct heat, the rate of heat conduction) in steel is much higher than that of wood. Therefore, per unit of time, the steel plate will have more time to “take” heat energy from the heated part than if this part is put on a tree, and thus it will cool faster.