DTFT frequency range
The DTFT is independent of Nyquist sampling. The DTFT is applied to a sequence of numbers regardless of any analog waveform that may be sampled. Economic sequences are often inherently discrete time sequences.  The DTFT applies to aliased sequences as well.

The summation is in general over $n$ from $-\infty$ to $\infty$.  The summation limits can be truncated to the nonzero range of $x[n]$.

The nominal convention as presented in Oppenheim and Schaefer, which they introduce prior to their chapter on Nyquist sampling, is that the DTFT is a continuous valued, periodic function defined uniquely on $-\pi$ to $\pi$ which has a length of $2\pi$. 

For real valued sequences, the interval $-\pi$ to $0$ is conjugate mirrored of the spectrum over $0$ to $\pi$.

In general for complex valued sequences, the mirrored conjugate symmetry can’t be assumed.

$0$ to $\pi$ is only one half of a length $2\pi$ period. 

edit

$-\pi$ to 0 has an image at $\pi$ to $2\pi$ and $3\pi$ to $4\pi$ and ......

0 to $\pi$ has an image from $2\pi$ to $3\pi$ and $4\pi$ to $5\pi$ and ....

similarly for the negative frequencies.