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.
According to Wikipedia: "The tip of the propeller on many early aircraft may reach supersonic speeds, producing a noticeable buzz that differentiates such aircraft. This is particularly noticeable on the Stearman, and noticeable on the North American T-6 Texan when it enters a sharp-breaking turn. This is undesirable, as the transonic air movement creates disruptive shock waves and turbulence. It is due to these effects that propellers are known to suffer from dramatically decreased performance as they approach the speed of sound. It is easy to demonstrate that the power needed to improve performance is so great that the weight of the required engine grows faster than the power output of the propeller can compensate. This problem was one that led to early research into jet engines..."
So basically the performance drops so badly that you better switch to jet engines. In fact when designing the blades, a max tip Mach of ~.9 is a hard limit of design. The text continues to say that nevertheless some propeller aircrafts were able to approach speed of sound in a dive.
No propeller-driven aircraft has ever exceeded Mach 1. The Brits did high-Mach dive tests in Spits out to high Mach numbers but never to Mach 1. In the US, Herb Fisher did high-Mach dive testing in a modified P47 and never reached Mach 1.
The problem is propeller over-speed and high drag rise on the propeller disk. It simply can't be done.