Can propeller driven aircraft sustain speeds at or above Mach 1? Has any propeller driven aircraft ever achieved and sustained supersonic speed? If not, why can't propeller airplanes achieve that speed?
Supersonic level flight?  No.

The fastest propeller-driven aircraft is either the turboprop-powered TU-114 with a measured top speed of Mach 0.73 or the turboprop-powered XF-84H "Thunderscreech" with a design top speed of Mach 0.9 and an actual top speed of either Mach 0.83 or Mach 0.7 (sources disagree).  The latter is an indication of why supersonic propeller-driven aircraft are unlikely: the continuous sonic boom coming off the propeller tips made the Thunderscreech the loudest airplane in the world.

The record for piston-engine aircraft is slightly lower, at Mach 0.71.