Since $V_\kappa\models{\rm ZFC}$, it satisfies that there are ${\rm ORD}$-many $\beth$-fixed points. But each of those must really be a $\beth$-fixed point because every $V_\alpha$ of $V_\kappa$ is a true $V_\alpha$ of $V$. Thus, there are $\kappa$-many $\beth$-fixed points below $\kappa$.

If $\kappa$ has cofinality $\omega_1$, then the worldly cardinals are unbounded in $\kappa$. Suppose $\xi<\kappa$. Let $V_{\alpha_0}\prec_{\Sigma_1} V_\kappa$ such that $\alpha_0>\xi$. Now inductively, choose $V_{\alpha_{n+1}}\prec_{\Sigma_{n+1}}V_\kappa$. Let $V_\alpha$ be the union of the $V_{\alpha_n}$. It follows that $V_\alpha\models{\rm ZFC}$ and since the cofinality of $\kappa$ is $\omega_1$ it follows that $\alpha<\kappa$.