A bathroom exhaust fan should produce 4 air exchanges per hour. Your bathroom is 7 x 7 which is 49sq.ft Assuming an 8' ceiling the total volume would be 49 x 8 = 392cu.ft We want four complete air exchanges per hour so that would be 392cu.ft x 4 = 1568cu.ft per hour. There are 60 minutes in an hour so we divide 1568 by 60min and it equals 26.13cfm. You currently have a 50cfm fan, which is twice the size you really require yet your having problems with the mirror fogging up. I suspect your problem is not the fan. Nature abhors a vacuum therefore a fan cannot push air out of a room unless there is provision for an equal amount of air to enter the room. You can test the theory by running the shower and leaving the bathroom door slightly ajar to allow more air to enter the room. Generally they make a slightly higer gap under the bottom of a bathroom door to allow more space for makeup air to enter the room but in some circumstances that does not provide enough makeup air. The solution then is to install a duct through the wall to an adjacent room and put a grille on either side. That will allow the additional air.

You should have continuity between the two plug leads. If you don't then the windings are open. You need a new motor.

How about just building a shelf somewhere -- recessed, protruding or whatever you like -- with all the wires you could conceivably want available at that shelf? You'd want a 110v outlet to power whatever device you'd use there. You'd want wires going to the speakers (RCA jack?) and you might want a network cable in case you decide wireless isn't the greatest way to get the music from your PC to the bathroom. Hide those three things behind some sort of aesthetically pleasing facade and you'd probably be okay.

Yes you can though sometimes it's a bit rough to pull an additional conductor through a conduit that already has stuff in it. They make double switches that fit where one switch goes. The questions are: 1. Can the circuit handle it. Most likely yes, the heater+light rarely exceeds 1200 Watts. Of course, you'll have to decide what else is on that circuit (not really so much a code factor than a nuisance factor). If it's on the same circuit as the receptacles in the bathroom, anybody with teenager daughters knows that one of those monster blow driers can trip a circuit. 2. Can you shoehorn the unit you want into the space your existing fan is in and get the exhaust duct to line up.