For example, when calculating an average of angle $\pi$ and $-\pi$, just adding them up and divide with 2 is 0, but since they are the same angle, shouldn't it be $\pi$ or $-\pi$? (180 deg)
Do I always have to use this fomula when adding, subtracting, averaging an angle?
Guys, why everyone is commenting about the equality of plus and minus?
Using the average method in the link, the average of angle $\pi$ and $-\pi$ is $\pi$!
I want to know what is a correct way of adding, subtracting, averaging an angle
You should think of angles as representing travel around the unit circle, starting at $(1,0)$ on the $x$-axis. Then $\pi$ is a counterclockwise rotation half way around and $-\pi$ is a clockwise rotation by the same amount. In this sense they do in fact average to $0$, even though both rotations end at the same point $(-1,0)$.
Edit in response to comment.
The formula in the link calculates (in one sense) the average of the final positions of the rotations corresponding to two angles.
Whether you want the "average angle" to mean "average amount of rotation" or "average final position" depends on how you plan to use the average in an application. The link points out that on a clock, so for things like time of day, you want the average of the final position.
So the answer to this part of the question:
is no. You use this method sometimes.