Bathroom light switch, fan switch & GFI outlet under same wall cover plate
First replace your GFCI outlet. It is definitely malfunctioning. (As a side, I would also recommend that you replace both switches, especially if they are more than 10 years old.
Check to see if your hair dryer still causes the breaker to fail. If not, then you are good to go. If the breaker does fail then a) it is overloaded or b) it needs to be replaced.
Determine all the "loads," i.e. things plugged in or turned on that are controlled by the breaker. You do this by adding all the watt ratings and dividing the sum by the voltage in the circuit. (If you don't understand what I'm saying here, then you should really call an electrician.)
If your load answer is greater than the rating of the breaker, then you have too many things connected to that breaker's circuit. See if you can plug some of those items into outlets on another circuit breaker. If not, can you turn some off while you use your hair dryer?
If the load is less than the breaker rating then you need to replace the breaker. This may be something you would want a professional to do.
Disclaimer: I am a home repair specialist with an engineering degree. I have a good amount of experience with home electrical systems. However, I am not a licensed electrician and do not claim to be.