2014 Versa S Sedan 5 Speed: Car Runs When Shut Off and Reverse is Engaged & Tail Light Issues

My 2014 Nissan Versa S 5-speed sedan sometimes continues to run when the car is in reverse and I turn off the ignition. When this happens, the car shuts off as soon as I move the gear lever out of reverse. Also, when this happens, the right rear tail light acts funny: the brake light seems not to work and the 3rd brake light also does not. Furthermore, when I engage the right turn signal when all of this is happening, it flashes rapidly as if the turn signal bulb were burned out.

I don't know where to start on this...do you think it's a fault in the tail light or a fault somewhere else? I took apart the tail light and the bulbs look ok; no corrosion, either.