The taillight housing has two bolts that must be undone to remove it to change the lights - (seen in the pic below) remove the bolts and then swing the housing towards the outside edge of the car as there are two pins on the inside body edge of the housing that hold it to the body there.
The cig lighter is just a simple system so assuming the fuse is good (5 Amp, #32 in the inside fuse block) then I would suspect fried wiring to the lighter itself or maybe a shorted out lighter element? Are all the other things attached to the cig lighter fuse still working? (clock, ECT transmission switch, glove box light, etc.)