I don't believe 300.3(B)(3) would include the words cable and cords if it only applied to individual conductors. NM and plastic boxes would comply with 300.20(B) because it says where, and since we don't have any "where" single conductors pass through ferrous metals, we are by default already in compliance.

I didn't really read the subject line. Here's another interpretation: If you are trying to control 2 lights (both going on/off at the same time) from 4 different places, then you don't just use 3-way switches. You need 4 way switches too. I think it needs two 3-ways and two 4-ways. I'm sure you can google it to get a proper wiring diagram.

I'm a total amateur but it does seem a bit unusual for SWITCHES to all have two sets of wires. Usually power goes to the device being switched, then down and back to the switch, so there is only one piece of Romex coming into the switch. I doubt disconnecting one will be a good idea, it's that way for some reason. You can verify whether it's actually two hot wires, or as randy said above, the second is actually carrying power to down the line. Shut off the breaker, disconnect one set of wires from the switch so they are not touching the other set (or anything else!), turn the power back on and check the voltage on the two sets of wires. I suspect you will find one set is hot and the other is not. In which case, don't mess with what isn't broken.

I've done a couple of these, although it's been quite a few years. The ones I worked on (Nutone, which I think is related to Braun) used the same blower for both vent and heat. They had a solenoid-controlled damper that redirected the fan output from the duct to the room outlet. The VENT switch turned on the fan. The HEAT switch turned on the fan, and powered the solenoid and heating element. Clever! Since then I've run into a different brand (the name of which I can't recall now) that had totally separate units for exhaust vent and heater. If it's the former, the wiring for that would indeed be confusing to someone with (how do I say this kindly?) a rushed or inadequate education in basic electrical principles. The wiring diagrams posted in this thread are pretty clear as to how the unit is supposed to be connected, though I don't get why the PDF that you link shows wiring only for the switch and not for the fan unit. I understand that you have a reason not to get another electrician, but to be blunt, I'm afraid that, from what you describe, this guy isn't going to be able to handle it. You know what you need to do, you just have to find a way to get around whatever is keeping you from hiring the right person for the job. PS - the switch they gave you is pretty slick. I sure hope you can find another one similar to it 15-20 years from now when it quits working. Good luck!