Constant current LED driver using opamp
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. The redrawn schematic. This shows the feedback arrangement more clearly.


  What's the formula for calculating R1?


R1 and C1 are creating a low-pass filter between the base and emitter to help prevent oscillations.


  What's the formula to calculate the required low-pass filter capacitor?


$$ f_{cut-off} = \frac {1}{2 \pi RC} $$


  Is this how we calculate the current: output current = Vref / R2?


Yes. The voltage will rise on the output until V- = V+. This will occur when \$ I = \frac {V_{ref}}{R_2} \$.


  How reference voltage affects the circuit (decreasing Vref to 2.5V)?


The current will be affected. You need to reduce it anyway. See below.




As presented the circuit will try to pass 3.3/1 = 3.3 A. 
The emitter of the transistor will be at 3.3 V. That means that the collector will be > 3.3 V.
If 3.3 A was able to flow you would have a maximum of 1.7 V available for your LED and it won't light much, if at all, unless it's an infrared type (as it has the lowest Vf). In practice (as pointed out by @What below) the current would level off at a much lower value but more than enough to light an LED.








simulate this circuit

Figure 2. An adjustable current source. Any pot in the 1k to 100k range should work fine.

Drop the reference voltage to 1 V (for easy maths) and you should get it to work.