The buzzer is not likely to be related to the alternator function. There is a small switch inside the ignition switch that sets it off when you put the keys in there. Your rear dome light is most likely to be a problem at the light fixture itself, they get flakey with age. The light may not be connecting, or the wires to it may be corroded at the fixture. The wiring in the dash to the swtich, or the switch itself could be bad. Usually it is just the plug in connection that is loose there and may need to be pushed in tighter. You will need to pry that panel off to get to it.