How exactly does elasticity relate to slope?
The two demand functions $D_1(p),D_2(p)$ cross at the point $(Q,p)$. Their respective elasticities at price $p$ are
\begin{align*}
\epsilon_1(p) &amp; = \frac{\text{d}D_1(p)}{\text{d}p}\frac{p}{D_1(p)} \\
\\
\epsilon_2(p) &amp; = \frac{\text{d}D_2(p)}{\text{d}p}\frac{p}{D_2(p)}.
\end{align*}
However since both function cross at the point $(Q,p)$ we know that
$$
D_1(p) = D_2(p) = Q.
$$
But then
$$
\frac{p}{D_1(p)} = \frac{p}{Q} = \frac{p}{D_2(p)}.
$$
Meaning the only difference between their elasticities is $\text{d}D_i(p)/\text{d}p$, which is their slopes.

As for your 1. question, the conditions are not clear. Is the 'flatter' curve only 'flatter' locally, or for every price $p$? If you only mean locally, then no, the statement is only valid in the intersection point $(Q,p)$.