If it was my house, I would put a porch over those doors. You can already see the rot beginning in the picture, and it's only going to get worse Every exterior door needs a porch. Putting a light on the ceiling of the porch also solves all your other issues about removing siding for a light.
The easy, simple way to do a light is in the soffit. Do you have a soffit?
IF you are going to go for a wall mount light, like a sconce, I would cut an opening in the siding big enough for a junction box and then drill a half inch hole into the top plates. To get the wire through the top plates, tape a thin wire to the end of the drill bit that's sticking above the top plates. Then pull the bit out and tape the small wire to your romex and pull it back up into the attic. I've also used PEX tubing to get Romex through insulation where I wanted it to go. The PEX is stiff enough to get through all the insualtion, and the Romex slides through the inside of the PEX. You just leave the PEX in the wall after the wire is threaded to where you want it.
If you want a light switch, hopefully it's on the inside wall where the exterior light is located so you have a short and simple run. With LED's burning so little energy, going with a dusk to dawn light and not having a light switch is something I've been doing more and more. I think I have seven of them on every night now and haven't noticed an increase in my electric bill.
It's probably not legal, but another way of doing this is to remove the trim around the door and running the wire through that space.