They apparently taxi with the mixture full rich causing the plugs to foul (mixture error one).
I just checked. There is no recommendation or instruction to lean the mixture during taxi in the 1988 PA28-161 Cadet POH. The only thing regarding this is "avoid prolonged idling at low RPM to prevent plug fouling".
The pilot and CFI, also apparently, didn't know how to clear the plugs by leaning the mixture at a higher RPM (mixture error two).
Again not a procedure that's listed in the POH.
Control of mixture during hot start (mixture error three).
POH has a hot start procedure. All it specifies is "mixture full rich".
I agree with Mad Jock that the cause is most likely overpriming, NOT incorrect use of the mixture - at least not "incorrect" according to the POH. Furthermore you don't give sufficient evidence that they executed the "induction fire" emergency procedure incorrectly. So I wouldn't be so quick to pile all the blame on the CFI or the education system that he/she grew up in. I'd rather wait for the results of a formal investigation of this incident - one that gives the CFI and the pilot a chance to comment as well, and incorporates the results of a technical survey of the aircraft.