How do I replace this odd outdoor outlet and light setup?
If this was my home, I would do the following:


I'd flip  the appropriate breaker, take everything out and disconnect it.
I'd take off a piece of siding and get it color matched so I could fill the gap. That would either be with buying siding that matched; or buying white siding with the appropriate color matched paint to fill it in.
I would take off a piece of interior drywall to give me access to the studs and add the electrical box and the light the correct way, with all wires running through the interior wall.
I'd patch the drywall and repaint the section of the room (Again color matching off of a section of the wall that I removed in the process).


You could fish the wall; and I've done it. I used to install coax cable for several years and have probably done 500+ wall fishes. However, for a complex and messed up setup like this it is almost always faster to cut the drywall, patch and paint. 

I would also REALLY like to see how the interior wiring of that thing was done on the parts you can't see. If someone was that creative getting the outlets there, I imagine there could be an interesting story behind the wall. Most interesting stories I don't want to find, but electrical can burn things down.. and therefore I most definitely would want to know.