I can't understand the proof of Lemma3.3 in Stability of genus 5 canonical curves.

Let $C$ be a complete intersection of three quadrics in $\mathbb{P}^4$ and let $\Lambda$ be the net of quadrics containing $C$. If the discriminant $\Delta(C)$ is reduced plane curve, then $C$ lies on a smooth quartic del Pezzo surface, defined by any pencil $l\subset \Lambda$ transverse to $\Delta(C)$. Why?

Conversely, if $C$ lies on a smooth quartic del Pezzo $P$ in $\mathbb{P}^4$, then the pencil of quadrics containing $P$ has a reduced discriminanat. Why?

Does only smooth quartic del Pezzo surface(among quartic surfaces in $\mathbb{P}^4$) have a reduced discriminant?