No, I don't have a meter - I'm doing this all from memories of my high school electronics classes! I'll get one, though.
I checked again, the labels over the lamps definitely say 12V d.c 2 W (just like that!) so I think I am ready to move on to the next step, which is the electronic parts...
Yes, that was exactly what I was planning. In fact, the fan is plugged into a conventional 110 socket built into the wall for the purpose.
A sticker above the lamps says 2W 12V (do I assume this is DC?). The idea would be to get a project box, mount a relay that handles a 12VDC control...
Sorry for being so low-tech - I have a kitchen range exhaust fan with lights built into it which are just ridiculous. For some reason the manufacturer thought that 2 X 2 Watt LED lamps were adequate lighting. They barely qualify as night lights. No wonder this super expensive-looking hood was so...