Have a look at the old bulb first and test it across the battery, if the bulb is blackened then it probably needs renewing, or it could be that the door switches are a bit sticky.
You are correct about the little square switch under the dash, it turns the speedo lights off if you just want parking lights, sounds like you need to check the wiring to it.
Do all the lights and horn work OK? if not check the fuse.
The heater is operated by a push/pull knob under the dash (push to open the valve on the engine) , you can just see part of it on the right of the picture, the toggle switch under the heater switches the fan on, I think you can just see the positions of screen, car and off on the flap control on the heater box.