I don't believe the rear door opening turns the dome light on, in my W108 there is a switch on the dashboard to turn it on, near the headlight switch. Its not connected to the door.

The headlights may be wired incorrectly or blown if the floor shift light comes on and they dont. When I first got the car the indicators wouldnt work due to a build of between the connectors in the acctually headlights. If its been sitting for a while they can get very clogged. Do you have American or European headlights? Try and clean up all the wires.

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1971 280SE - The S Class
1974 280CE - Sterling
1988 300E - Stanley