Question: The GFI in our bathroom was tripping somewhat frequently when my wife used the blow dryer. Other than the vanity light there were not other things operating on this circuit. If the GFI did not trip, the plug to the hair dryer would be quite hot to the touch. To my limited knowledge, a danger sign.

After a ridiculous quote by an electrician to diagnosis the problem, I decided to watch some you tube videos. I bought a GFI tester and the outlet was hooked up correctly. However, I noticed that the circuit had a 20a breaker, but the GFI was 15a. So first step, I replaced it with a new 20a GFI. After testing all symptoms were gone. No hot hair dryer plug and no tripping.

I checked the other GFIs despite them working correctly. All were 15a GFIs but on a 20a circuit. One was the kitchen which get used very heavily. I went ahead and replaced them all, just to be safe but it left me with some questions:

1. How bad is it to have a 15a GFI on a 20a circuit?
2. Do you think that the problem in our bathroom was just a bad GFI? Related to the mismatch in circuit and outlet rating? Or something else?

Answer: 
My understanding is that you can have multiple 15 amp outlets on a 20 Amp circuit but not just one. GFCI's trip on a fault, not an overload so the existing GFCI was probably going bad. Hair dryers can easily use 15 amps and that would heat up the plug a bit. If the GFCI outlet was old and had been used a lot, the plug might not have been as tight as it should be but not enough to arc but would heat up more.
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> However, I noticed that the circuit had a 20a breaker, but the GFI was 15a. So first step, I replaced it with a new 20a GFI.  
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> How bad is it to have a 15a GFI on a 20a circuit?
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> 

It's not bad at all and is totally legal. There was never a problem here; you reacted to this over nothing.

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> Do you think that the problem in our bathroom was just a bad GFI? Related to the mismatch in circuit and outlet rating? Or something else?
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> 
> 

Fair chance there was nothing wrong with the old GFCI. The heating of the socket is the clue.

You fixed the problem "by accident" in the course of swapping the GFCI.

What you actually had was a bad connection on the "Line" terminals. Most likely, a screw was not torqued correctly, and the connection was loose. Loose connections cause *series arcing* and that creates a great deal of heat. Series arcing can also cause fires, which is why they have AFCI (Arc Fault) breakers now to catch these faults.

This is why NEC now requires torque screwdrivers to set torque wherever a torque is specified.

If that was an AFCI receptacle (e.g. an AFCI/GFCI dual-mode) then it may have been detecting this arcing. Because of the way the signal travels, an AFCI receptacle can "hear" a series arc fault in its supply wires.