Why is phase range between $-\pi$ and $+\pi$ (instead of $0$ and $2\pi$)?
It is just a convention, but it is useful in some cases. For example, the phase of the DFT of a real discrete-time signal is odd only if the angles are expressed in the range $[-\pi, \pi)$. Sometimes you just have to adapt to the convention used by your tools -- for example, MATLAB functions like angle and atan2 return angles in $[-\pi, \pi)$.
Note that the frequency of a discrete-time signal, measured in radians per sample, is in the range $(-\pi, \pi]$, but in this case it has a different interpretation.