I am use to homing the machine by hitting Home and then the axis will go to the home switch, stop, go backwards slowly, then go forwards slowly to hit switch again.  This makes an accurate home limit.  

My question:  is this function controlled by Mach3 or by the machine board?  In other words, does Mach3 give the board commands to do this (move fwd, move backward, move fwd again) or is it automatically done by the board when Mach3 gives it the home command?

AFIK, which isn't much it's all done by Mach.

By your description of the sequence it is your controller that is doing it.
Hood

AFAIK, homing is one of the "real-time" actions like limits or probing etc. so HAS to be done by the controller.

FWIW, your description does not match the way homing is "normally" done.

Actually I suspect your description of the homing sequence is incorrect because if it worked as you say it does then you wouldn't be able to share limit and home switches.