led operating voltage
The most important thing to understand is that LEDs (if considered simplistically) drop a constant voltage of somewhere between 0.7 - 5 volts. 
In the datasheet this is shown as the forward voltage as 1.9 for the red and 3.3 for the green and blue leds. 

They also don't offer any resistance. So it does not matter what voltage you apply to the LED so long as you limit the current. So if you were to apply 1000V to an LED it would drop say 5 volts across the LED and 995 volts throughout the rest of the circuit (so long as you limit the current and don't blow up the LED). 

Your calculation is not quite correct because you did not take the forward voltage of the LEDs into account. So for example, if you want 20 mA through the red LED you need to go:

$$R = \frac{12 - 1.9}{0.02} = 505\Omega$$

But often LEDs are driven by current sources rather than limited by a resistor which is probably what your LED controller is doing but you haven't provided a link for that.