Here's a little Youtube video of the real D-EBFW, in the initial scene particularly dismantled for servicing.
Later you can watch the Bad Nauheimer guys unfolding the wings, starting the engine and taking off.
no, it's a metal variable pitch Messerschmitt Me P-7 propeller.
By the time the video was taken it had some painted stripe on the "leading edge" that could be confused for wood.
IIRC they got the metal prop in 2009, before that they had a fixed wooden prop.
Most of the civil prewar Taifuns had indeed wooden props, the majority of the Luftwaffe Taifuns had metal props.