BJT switch vs amplifier mode
Ah, the joys of finding a random circuit on the Internet that happens to have a slick presentation. This is actually a very poor design for several reasons.

First of all, the author should have put a resistor between D10 and the capacitor, which would have allowed him to use a much smaller capacitor to get the cutoff frequency he needed. The VEE pin of an LCD requires only a tiny amount of current. As it is, he's relying on the output impedance of the Arduino pin to limit the current into/out of the capacitor, which is very poor practice.

Secondly, the transistor is being used in a common-emitter mode, not emitter-follower. Using two resistors the way he does doesn't make much sense.


  Why isn't a simple capacitor is used in the brightness pin, similarly to the contrast pin?


Two reasons:


The Arduino pin by itself can't handle the current required by the backlight. The transistor provides the necessary current gain.
In this case, the goal isn't to turn the PWM signal into a DC level, but instead, to use it to turn the backlight LED on and off rapidly to change the apparent brightness.



  In the emitter-follower, is the BJT operating in linear or saturation zones?


Like I said, this isn't an emitter-follower. Because of the resistor in the emitter leg, however, it's operating on the cusp between linear and saturated zones.


  Why does the author discard the base emitter voltage (0.6V) in his calculations?


I assume that when you say "discard", you mean "ignore". Good question, although it would really be the collector-emitter voltage in this situation. If he was using a circuit configuration in which the transistor would definitely be in saturation, this voltage would be relatively small (about 0.3V), but still significant.

The circuit would be better if the emitter of the transistor were connected directly to ground, and the resistor R1 were placed in the path between Q1's collector and the LED- pin of the display.