Outside temp I can not help you with.. Possible it does not exist.
DOME light I may be able to help you with.. I will describe dome light operation on MY RV.
First the light is a double dome RV ceiling light, with a switch (3 position) this switch must be in one of the ON positions for the light to work (The positions are OFF ONE BOTH on mine)
Next since it is a "Dome light" just like the one on the ceiling (Dome) of my car you have to turn it on from the dash.. USUALLY it is the dash lamp dimmer to full bright and push past a detent to turn it on..
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times