Let $S$ be the "usual pinch point surface defined by $x^2t=y^2z$ in $\mathbb P^3$ and $T\subseteq \mathbb P^3$ and arbitrary general surface of degree $d-3\geq 2$. Let $X_0=S\cup T$. Note that then $\deg X_0=d$. Let $\ell\subseteq S$ be the double line and $H\subseteq \mathbb P^3$ a general surface of sufficiently high degree such that $\ell\subseteq H$ and let $C_0=H\cap X_0$.

Since $T$ is general, it is smooth and intersects $S$ transversally, so I think $X_0$ is even semi-smooth, but for sure slc. Now choose a smoothing of $X_0$ as a degree $d$ surface in $\mathbb P^3$, i.e., let $f:\mathscr X\to B$ be a family of degree $d$ surfaces in $\mathbb P^3$ parametrized by a smooth curve such that $X_0$ is a fiber of $f$ and the general fiber $X_\eta$ is smooth. Since $d\geq 5$, $\omega_{X_b}$ is an ample line bundle for every $b\in B$.

Now let $\mathscr C$ be the collection of curves $C_b=H\cap X_b$ for $b\in B$. The general member will be smooth and $C_0$ contains $\ell$ as an irreducible component. So this seems to answer your first question.

For the second, take the same example, but add a $\Delta$, i.e., let $\Gamma\subseteq \mathbb P^3$ be a high degree general surface and let $\Delta_b:=\Gamma\cap X_b$ and $\Delta\subseteq \mathscr X$ the preimage of $\Gamma$ in $\mathscr X$. Then $\Gamma$ and hence $\Delta$ intersects $\ell$ transversally at a point which is smooth on $C_0$ and since $\Gamma$ is ample, $\Delta\cap C_\eta\neq\emptyset$.

Let $S$ be the "usual" pinch point surface defined by $x^2t=y^2z$ in $\mathbb P^3$ and $T\subseteq \mathbb P^3$ an arbitrary general surface of degree $d-3\geq 2$. Let $X_0=S\cup T$. Note that then $\deg X_0=d$. Let $\ell\subseteq S$ be the double line and $H\subseteq \mathbb P^3$ a general surface of sufficiently high degree such that $\ell\subseteq H$ and let $C_0=H\cap X_0$.

Since $T$ is general, it is smooth and intersects $S$ transversally, so I think $X_0$ is even semi-smooth, but for sure slc. Now choose a smoothing of $X_0$ as a degree $d$ surface in $\mathbb P^3$, i.e., let $f:\mathscr X\to B$ be a family of degree $d$ surfaces in $\mathbb P^3$ parametrized by a smooth curve such that $X_0$ is a fiber of $f$ and the general fiber $X_\eta$ is smooth. Since $d\geq 5$, $\omega_{X_b}$ is an ample line bundle for every $b\in B$.

Now let $\mathscr C$ be the collection of curves $C_b=H\cap X_b$ for $b\in B$. The general member will be smooth and $C_0$ contains $\ell$ as an irreducible component. So this seems to answer your first question.

For the second, take the same example, but add a $\Delta$, i.e., let $\Gamma\subseteq \mathbb P^3$ be a high degree general surface and let $\Delta_b:=\Gamma\cap X_b$ and $\Delta\subseteq \mathscr X$ the preimage of $\Gamma$ in $\mathscr X$. Then $\Gamma$ and hence $\Delta$ intersects $\ell$ transversally at a point which is smooth on $C_0$ and since $\Gamma$ is ample, $\Delta\cap C_\eta\neq\emptyset$.