Electronic – AL8860 LED driver does not output required constant current

1. Yes, you should care. The BCR421 will try to maintain a (roughly) constant current at the collector terminal, however if Vout drops too low then it will not be able to maintain the constant current, as you can see from the graphs you posted. What will happen is that the output will saturate at a lower current than you want. To find out if this will happen, assume that the maximum desired current is flowing through the LED. Subtract the maximum possible LED voltage (which will occur at the maximum current) from the minimum possible supply voltage and compare with the dropout voltage of the BCR421 at that current.

   So the LED drops 1.35V typical at 100mA (1.6V maximum). Your supply voltage is 3.3V nominal, let's dock 5% for tolerance to give 3.13V. That gives the chip 1.54V to work with. The parameter VSmin indicates that typically 1.4V is enough at 25°C, and the graphs you posted indicate that above about 1.1V it's fairly constant current. So it's likely okay unless you have some extreme operating conditions.

2. The current through the external resistor when it is regulating correctly will be approximately Iout - 10mA. The internal resistor conducts the 10mA. It will be slightly higher because of base current (maybe 0.5%), but that's immaterial in this situation, given the +/-10% tolerance on the 10mA current. As @jippie points out in a comment, 100mA through a pot wiper is probably not a good idea, depending on the pot construction (okay for a wire-wound, but they are expensive). Minimum resistance may also be a problem. It's generally recommended not to put much current through pot wipers- they work better as voltage dividers than rheostats.

3. This circuit maintains a fairly constant current regardless of changes in the LED forward voltage (or differences from unit-to-unit when first assembled) and regardless of changes in supply voltage.

You have not mentioned the color of the LEDs. The typical white LEDs that make up modules like yours would take more than 3V to reach maximum rated power. I would say around 3.5V could be a common number.

For a 3.5V white LED, probably around 20% of the voltage drop is due to ESR. A LED (of the same base color) requiring lower voltage and therefore has a corresponding lower ESR would be more efficient.

So there is an unavoidable resistor in series with every LEDs already. If you use the number 12V, 0.6A, 20% ESR then the ESR = 12 * 20% / 0.6A = 4 ohms. These are guesses and may not reflect what you have. But say if these guesses are exactly right, then you would get 1A at 13.6V.

(The simple ESR model breaks down badly at low current).

Rereading what I wrote, I was assuming 4 x 3V. But there are only 3 LEDs in series, so the numbers do not add up.