Body Control Module in front of fuse box in engine compartment, remove it and clean all contacts with emery cloth. Then using needle nose pliers twist all contacts just slightly all in the same direction. Reinstall and this should work.
Actually I would plug in something else that works on 12 volt to verify it is not the outlet. I really think this is going to be your charging cord for your phone and not your lighter port. If that checks out then your fuse panel will be on the lower right or left side of your car by your feet behind a pull off panel.
Could be both. The switch selects what section of the resistor gets power. If the switch is good then the resistor is bad. If the resistor isn't open or burned, then suspect the switch.
In my van.. I have the front heater switch n a separate switch for the rear. They are separate units though. One fan runs the front and a different fan runs the rear.
Probably did something wrong installing the starter. I would check make sure you got the right one. Then check to make sure you hooked up the connections right and that its grounded properly