Jeff (my parents are from Wilmette and Evanston), Big Chainring (I'm in the Other Washington, up the road apiece from you), thanks for the thoughts. I'm learning all kinds of new stuff!
Chainring - the RD cage is pegged even when I'm on the middle chainring and the largest rear sprocket. When I'm on the largest chainring I can't even get it on the 26t or 36t (largest two) sprockets. On the other hand, the chain droops (e.g. hits the chainstay) when I backpedal while on the smallest chainring and *any* of the rear sprockets.
Jeff - I'm amazed you can tell so much from those pictures. I'll definitely replace the gear and brake cables / housing. That's good to know about the rear wheel - I didn't even know what rear dropout adjusters were until I looked them up after reading your post (and you're correct, there are holes for them, but no adjuster screws present). I already fixed the handlebars after this picture was taken. These are good places to start - I'll fix the wheel and cables and see how things look at that point. Maybe it'll all just start working! (Ok, not likely...)