"Shuffle off this mortal COIL" (Shakespeare)

I believe this word "Coil" could be a misheard misnomer.
The correct phrase might be "mortal CAUL" which one indeed "shuffles off". The caul was given great importance in the past.

No, it is certainly not misheard. Shakespeare used the word coil numerous times in his plays. It has been in the language since at least 1570 and means fuss or disturbance.