1 - After pressing "subscribe" it showed up on the list but vanished after restarting the game and I couldn't get it back.

I got it working by copying the mod folder (834405373) from:

\steamapps\workshop\content\270880\834405373

To:

\Documents\American Truck Simulator\mod\

It showed up on the list, I added it, and saw it when I went to a repair shop in the passenger side license plate section. I didn't check the driver side license plate section to see if it was also there.

2 - When I added a cabin light, it asked for text "Enter co-driver's nickname". I left the field blank, clicked "Apply" (as if I was buying a blank plate) and it added the light.

3 - Tonight, I reloaded a previous save game and went to a repair shop and didn't see the option to buy the cabin lights in the passenger side plate section anymore. The light does still work in my current vehicle, though (I haven't gone to a repair shop to see if it shows up in my current vehicle).

I found and downloaded another copy of the mod from:
http://www.modsats.com/cabin-lights-ats-1-5-x-mod/

It may be a more recent version since it was uploaded in Feb while the Steam version was uploaded in Jan. The modsats version does show the option to buy the cabin lights (but still asks for plate text).

I don't use Windows 10, but maybe someone who's using it can test this version of the mod by putting it in this folder:

\Documents\American Truck Simulator\mod\