What is the relation between throttle position and engine thrust for a turbofan?
Check out this Nasa powerpoint it provides a nice overview of some of the basics and how some of the systems are set up. 


  Does the auto-throttle send its command directly to the EEC or to the
  mechanical throttle?


This may vary from design to design but the Boeing 777 uses the ARNIC standard bus to send control signals to servos that physically move the throttle levers. You can read up on the ARNIC 429 data bus here. 


  Are the throttle levers customized/calibrated for an engine model, or
  are their output standardized (assuming the output is an electric
  signal). Does the electronic engine control apply further adjustment
  to the throttle input?


It seems that Rockwell Collins makes a unit that sends signals to the FADEC system instead of mechanically actuating anything.

With that in mind Im sure there are older designs out there that are closer to or a complete direct linkage from lever to engine.  


  Is this correction transmitted back to the throttle lever?


I dont think so, it seems that when say full throttle is applied the throttle control will remain at full and the FADEC unit will take care of spooling it up as need be. Likewise if an engine suddenly cuts out or fuel flow becomes restricted as far as I know the throttles will remain in place. Auto throttle units will however physically actuate the throttles as need be. As per the comments here it seems this is only the case in Boeing aircraft. Airbus throttles must be set to maximum setting auto-throttle is allowed to use.