The dash light won't light up and the climate control will go dark when the marker's or the headlights are on if the dimmer is burned out or disconnected. To test this one part just pull the dimmer out and jumper the white wire with black stripe to the white wire with green stripe.
As for the headlights, I'd start with testing the switch.
The headlights on our cars will turn off after the car is turned off if you open the door.
If this isn't happening, then I's skip the switch and head straight to the relay.