Does unit elasticity has to be at exactly the middle of the demand curve?
No, that is only true in the linear case. For a simple counterexample consider
$$
D(p) = 1 - \sqrt{p}.
$$
$$
\epsilon(p) = \frac{d D(p)}{dp} \cdot \frac{p}{D(p)} = -\frac{1}{2  \cdot \sqrt{p}} \cdot \frac{p}{1 - \sqrt{p}} = \frac{1}{2}  \cdot \frac{\sqrt{p}}{1 - \sqrt{p}}
$$
\begin{eqnarray*}
\epsilon(p) &amp; = &amp; 1 \\
\\
\frac{1}{2}  \cdot \frac{\sqrt{p}}{1 - \sqrt{p}} &amp; = &amp; 1 \\
\\
\sqrt{p} &amp; = &amp; 2 - 2 \cdot \sqrt{p}  \\
\\
\sqrt{p} &amp; = &amp; \frac{2}{3} \\
\\
p &amp; = &amp; \frac{4}{9}.
\end{eqnarray*}
This is not in the middle of the demand curve in any sense.