Why does connecting batteries in series result in doubled current sourcing while shorting?
I think this is due to how you are measuring the current.  Your results all make sense if the resistance of your current meter is higher than the internal resistance of the battery at the measured current.

Think of the limiting case where the batteries are pure voltage source (no resistance) and you put a fixed resistance accross them (your current meter in this case).  Under these conditions, the current would be proportional to the total voltage, and you'd get pretty much what you measured.

The other limiting case is where each battery has a fixed finite internal resistance.  This can be thought of as a resistance in series with a ideal battery.  In this example, the "short" is truly a short of 0 Ohms.  Now the short circuit current of any one battery is fixed, which is the battery voltage divided by its internal resistance.  In this case adding any number of batteries in series won't change the short circuit current, but putting then in parallel multiplies the current by the number of batteries.

From your measurements, I think what is going on is close to case 1, not case 2.