For how long can a single-engine turboprop safely run the engine at idle? Can a single engine turboprop (such as a Piper Meridian or TBM) allow idling long enough to drop off a passenger or package without shutting down the engine? If so, would the prop be feathered but still spinning? Would idling with the prop not spinning (if possible) hurt the engine?
It's not stressful at all for a turbine engine to run at idle, you could run it until the fuel runs out. The prop is not feathered (in line with the air stream, pitch 90°) but will be at ground fine, in line with the direction of blade turn, pitch angle 0°.

There is usually no way to disconnect a propeller from a single spool turbine engine: if the engine runs, the propeller turns.