Why is X(0) the DC component
Follows from the DFT definition. It's defined as

\begin{equation}
X(k) = \sum_{n=0}^{N-1} x(n) e^{-j2\pi \frac{kn}{N}}
\end{equation}

So $X(0)$ is

\begin{equation}
X(0) = \sum_{n=0}^{N-1} x(n) e^{-j2\pi \frac{0 \cdot n}{N}}
\end{equation}

Having $k=0$ gives $e^0=1$ all the time so that

\begin{equation}
X(0) = \sum_{n=0}^{N-1} x(n) 1
\end{equation}

Comparing this to the average

\begin{equation}
\overline{x} = \frac{1}{N} \sum_{n=0}^{N-1} x(n)
\end{equation}

shows that $X(0) = N \overline{x}$