Is it possible to cut off and/or grind down cable and downtube shifter bosses without harming the integrity of the frame? I might have a frame refinished but would really really like to get rid of these before new paint.

Yes. I did it by holding the frame and using a bench grinder. It takes practice to not slip and grind a big dig in the tube. That could ruin it. A dremel tool will do it.

Use a propane torch and a set of pliers. Then a hand file to smooth out the rest. Power tools and thin metal tubing don't usually mix.