Interpreting the differential resistance of a Zener diode
You use the 1 mA column if you are intending to run your zener at a current of around 1 mA. You use the 5 mA column if you are intending to run your zener at around 5 mA.
For the 12 V BZX84 diode in the link you supplied, the typical values are 50 Ω at 1 mA and 10 Ω at 5 mA. That's a huge performance improvement, at the cost of power consumption, if you run at 5 mA rather than 1 mA.
So when designing your circuit, you have a choice. If it's to be battery powered, then you probably want to sacrifice performance for power consumption. If it's mains powered, and you can afford the extra heat dissipation in the space you have, you can get better differential resistance by running at 5 mA rather than 1 mA.
You would never apply 16 V to a 12 V zener. Or if you did, and the power supply had the current capability, it would fail in microseconds. You would use a resistor between the supply and the zener to supply it with a well defined current. You would get 1 mA though a 4 kΩ resistor, and 5 mA through a 800 Ω resistor.
If you wanted to know the voltage to expect at 3 mA (with a 1.3 kΩ resistor), then you could have a go at interpolating the 1 mA and 5 mA voltages, using the curves of figure 6 to guide you. However, given the zener tolerance is around +/- 200 mV, and the temperature coefficient can be as high as 10 mV per K, there's little point in trying to be more accurate than 'about 12 V' when dealing with this class of component.
Figure 6 is also useful to answer your 'minimum current' question. It shows voltage versus reverse current for a number of diodes, the current plotted from 10 nA to 20 mA. The low voltage diodes &lt;= 10 V and the high voltage diodes &gt;= 18 V show a drop in voltage down at the 10 nA end. However, the intermediate voltage diodes, including the 12 V diode, show remarkably straight lines over the full current range. From this you can deduce that they need at least 10 nA to work. Of course their differential resistance will probably be very high at this current level.