I thought charging at that low of an amperage uses more energy than at 20-30A. Something to do with overhead but might be wrong.
The 5 amp setting does use more electric as mentioned on other posts I have read on the subject. Much of the time I choose 12, 16, 24, 30, 32 or 40. Part of the reason I have selected different charge rates is to check what 90% mileage range number occurs at the end of charge. Those charge rates also simulate what occurs with some of the J1772 chargers.
The most efficient and battery saving charge rate would be an excellent number to know. After reading the following, a higher rate is more efficient.
Searching the web, there are differences between different EVSE models:
http://energy.gov/sites/prod/files/2014/03/f13/vss096_francfort_2013_o.pdf
You definitely want 240 volt charging >2 kWh after reading page 9 here:
https://www.veic.org/docs/Transportation/20130320-EVT-NRA-Final-Report.pdf
Keep reading that report for temperature differences. The best efficiency is between 53 and 70 degrees.
One more link to share clearly shows the efficiency improvements at higher rates:
http://avt.inel.gov/pdf/fsev/SteadyStateLoadCharacterization2015Leaf.pdf
The question that remains is what amperage level at 240 AC voltage is best for the health of the battery. I called Tesla Motors and they did have some advice, but no charging specifics for the health of the battery, other than avoiding range charge above 90% unless needed and charging before zero miles remain. (IMHO, I attempt to have 50 miles of range at the next Supercharger, it can help to have a buffer of range in windy or cold weather.)
Since the Tesla vehicle will heat or cool the pack as needed for the health of the battery, Tesla did suggest to charge before the battery becomes cold in the winter to avoid a heating of the pack, drawing more electric. IMHO, charging at a lower rate when the battery is hot could help save electricity by not having the cooling fans running, however in the summer, setting a timed start for the morning when the battery pack is cooler should be best.
OT: This brings up one suggestion I have made to those who travel in the winter. Always Supercharge before an overnight stay since the battery is warmer and will take the charge faster. If you have freezing weather overnight and wait until the morning, you will have a longer wait at the Supercharger.