Current codes requires exhaust fans in bathrooms. While this does not affect your's, a fan is always a good idea. Fans must exhaust outside. If the bathroom is on an outside wall, or has an open attic above, installation is fairly simple. If located towards the interior of the house, it becomes more difficult, but not impossible. As to sizing, measure the square footage, multiply by 1.5, that should equal the CFM of the fan.
Ceiling fans and exhaust fans are two different things.
A ceiling fan, while one could be installed if the space permitted, is probably not the best solution for a hot bathroom. All it would accomplish is a means to move the hot moist air around. A ceiling or wall mounted exhaust fan would do a better job of getting rid of the heat and humidity, provided there is a source of cooler dryer make-up air, such as a door grill or large enough opening under the door.
I think the fist approach should be to find out why the bathroom is so hot in the first place and find a remedy. For instance, is there enough insulation between the ceiling and a hot attic above? Is the sun beating directly against a poorly insulated outer bathroom wall?
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