Spooner's music teacher? He's six feet under! (9)
I believe the answer is:
cockroach
'he's six feet under' is the definition.
(ie he has six feet underneath him)
I don't understand the remainder of the clue.
(Other definitions for cockroach that I've seen before include "See pest" , "Cork coach can be a pest" , "Household insect pest" , "Rock coach turns out to be an insect" , "So-called black beetle" .)