Can a plane fly backward if its propeller is in reverse?
Fly? No.
Wings generally only produce enough lift to keep an aircraft aloft when air is flowing over them in their intended direction. If you reverse the airflow over the wing (by moving backwards in the air, for example) the wing would no longer be producing the necessary lift, and the aircraft would "descend at an extremely rapid rate" until a normal airflow over the wings was restored. (That's the polite way of saying "the darn thing falls out of the sky like a rock!")

Move? Yes. At least on the ground.
This has been demonstrated on more than one occasion by Fat Albert, the C-130 that supports the Blue Angels.  While the propellers themselves do not reverse, the blade pitch is changed to produce reverse thrust (slowing the aircraft the same way thrust reversers do on a jet engine, and in this case reversing its direction of travel).



Note that there is one notable exception to the "airplanes can't fly backwards" rule, which the article Ethan linked to alludes to: If the wind velocity exceeds the aircraft's stall speed it's possible for an aircraft to "fly" at its minimum airspeed, but be moving backwards relative to the ground.
This is a bit of a cheat though: The airplane still thinks it's flying forward (the relative velocity of the wind over the wings is in the "normal" direction, and the airspeed is fast enough to create enough lift to sustain flight). It just happens the the airspeed includes a headwind component sufficient to give the aircraft a net "negative" groundspeed.
The hypothetical P-51 in your question would require sustained winds of about 83 knots to make this trick work (we generally call that a hurricane), but something like a Piper Cub can do it at much more reasonable wind speeds.