No. The only thing required to be gfci protected is a receptacle. I've never understood why light fixtures, fans, and other appliances do not required protection around so much water, but they do not. They should, however, be listed for damp locations.
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Deoends on code you are governed by. In USA, a GFCI outlet or a circuit controlled by a GFCI circuit breaker would be required.
Not if the GFCI breaker is supplying the circuit you are wanting to put the GFCI receptacle into.
There is likely an internal short circuit in the fan motor.
GFCI receptacle's must protect any outlet that is within 6 feet of a water source, outside the home, in a garage, on a deck, in a sunroom, in a bathroom, unfinished basements, kitchens, & crawl spaces.
Your question sounds simple but it isn't. There are two ways of wiring a light to a switch. One is to bring the hot and neutral wire to the switch box and from there run wires from the switch to the light. If this is the case you can install a GFCI on the circuit. The other way is to bring the hot and neutral wire to the light and from there run two wires to the switch and switch the light that way. In this case you cannot install a GFCI to in the switch box. <<>> The way you want the circuit to work has to be laid out as follows. Method one, the circuit that now controls the bathroom devices can be changed to a GFCI breaker. This is probably the quickest and cheapest, unless the job is new construction. Method two, if new construction use this method, as tearing walls open to get to the wiring will become costly. Where a supply circuit comes from the service distribution panel the first device on that circuit has to be a GFCI receptacle. This device has secondary terminals on it, that if connected every device downstream from it will also be protected. So if you supply the two way light switch that controls the vent fan and a bathroom light from the secondary terminals from the GFCI receptacle they will be protected.