Congrats on the new ride!
I will let others comment on the start/run problem....excpet to say that it could be the wiring that is connected to the neutral safety switch....but if that were the case it would be a problem with it starting and stopping, but not in drive I wouldnt think...the NSS (on auto trannies at least) prevents starting the car in any gear....only neutral or park......
The instrument panel is not backlit......(that concept was new to me too, so dont feel bad...LOL)......it is lit from the front, using several small bulbs behind blue/green plastic lenses built into the metal strip directly above the panel...there are 4 or 5 screws (I think) holding it in and it drops down.....
The headlight dimmer switch controls the intensity of the light, but even after cleaning my dimmer switch really well its still leass than impressive. There are a number of threads on this forum (search for dimmer or sintrument lights) about fixes/tweaks/mods for the instrument panel lights.....one is removing the plastic lenses altogether....as Chryco talks about in this thread....
http://www.cuda-challenger.com/cc/index.php?topic=60732.0For the dome light, check the bulb first but it could be that the pin switches in your door jambs are crusty/not working/missing......
The burned left rear signal could be causing that side signal to blink funny.......so check the bulb and if its not that, the socket etc. The grounds on my tail-lights were bad and I had all sorts of weird signal/light issues til I went over those.
Good luck and welcome!