= Solution
{c}
The collapse $\pi:H\to N$ is a <bijection>, so $|N|=|H|=\aleph_1$. Every $\alpha<\omega_1$ and every element of $\alpha$ belong to $H$. The <transitive collapse fixes transitive subsets> lemma therefore gives $\pi(\alpha)=\alpha$, so every ambient <countable ordinal> belongs to $N$.
Moreover, $V_\kappa$ correctly regards each such $\alpha$ as countable. An injection $\alpha\to\omega$ has rank below $\max(\alpha,\omega)+\omega<\kappa$, so a witness belongs to $V_\kappa$. Elementarity gives $H\models$ “$\alpha$ is a countable ordinal”. Transporting this statement across the collapse, which fixes $\alpha$, gives the same statement in $N$.
The collapse also fixes the element $\omega_1\in H$. No injection $\omega_1\to\omega$ exists externally, hence none belongs to $V_\kappa$; together with the countability witnesses below it, this shows that $V_\kappa$ correctly identifies $\omega_1$ as its first uncountable ordinal. Elementarity and the collapse preserve that identification. Thus the construction gives
$$
\boxed{|N|=\aleph_1,\quad\alpha\in N\text{ and }N\models\text{“}\alpha\text{ is countable”}\ (\alpha<\omega_1),\quad(\omega_1)^N=\omega_1.}
$$
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