Like Hank mentioned that stuff goes through the steering column. That's how it knows to have the brake light on solid (braking) or on and flashing (braking and turning).

I'd also look to make sure the bulb sockets are clean, grounded, and the wiring is in good condition.

As for the dome light, isn't there a separate fuse for that? My experience with the newer (offshore made) headlight switches, quality wise, isn't that great. I've had some right out of the box that have dead spots for my gauge light dimming. The dome light is just a switch position when you turn the knob all the way CCW. So the dome light problem has to be the switch, fuse, bulb, or wire. Once again, you might want to check the socket, and wiring. If I recall correctly the dome light has 12 volts constant and the ground is supplied through the door switches (when open) or the headlight switch. If that ground connection is bad you'll loose your dome light.