Confirmed at least with my '14 FFE and JuiceBox EVSE: 5A is the minimum current for both 120V and 240V. When I set current on the EVSE to 4A the car will not engage at either voltage. Set to 5A and it starts charging.
Interestingly, my FFE got pissed when I went from 240 to 120 without unplugging from the car. The charge ring was flashing quickly, and would not charge. I had to unplug and replug before it was happy again. When i switched back to 240 without unplugging from the car, my FFE did not complain.
Since I also have a model 3, it would be nice to charge both the model 3 and my FFE with the same EVSE without having to remove the tesla adapter. It's pretty low on my priority list, but if I ever get around to it I plan to build an arduino based EVSE with 2 of J1772 plugs. I am thinking of having 2 operating modes. Mode 1 would split the max current so I can charge both vehicles simultaneously (50A circuit so each car could get max 20A). Mode 2 would power Plug A first then power Plug B. If Plug A requests power it would interrupt Plug B. This should be fairly easy to implement as long as the arduino has enough IO.
FWIW you could charge at 6A @120V but that is only 720W. If it is cold out I'm not sure how much if any range you would add, as there are 900W of heaters in the coolant loop (3x 300W). Neglecting any power used to charge 12V battery, run pumps, heaters, etc and assuming 90% charge efficiency and 300Wh/mi energy consumption, you are looking at barely over 2 miles of range per hour of charging. 720Wx0.9/300Wh/mi=2.16mi/hr charging. Real world probably less than 2 mi/hr. So yeah... not really practical unless you have lots of time to kill.
2014 Blue Candy FFE
http://www.fuelly.com/car/ford/focus/2014/triangles/303811 (since this forum doesn't allow BBcode in sigs)