Voltage drops when turn on electric fan

Electric motor drag always causes voltage drop, more specifically an amperage draw increase.
Still your alternator should be able to overcome that, not that the motor getting that hot is ok.
It does sound like something isn't putting out like it should.

On the fan you really should have relays for the two speeds right by the fan with as short and as large a wire as you can stand (like 10ga) and use your switch wiring to activate the relays.
I'd bet those wires and or switches get hot, right?