How does $\mathsf{NP} \subset \mathsf{P}/\mathsf{poly}$ imply these two inclusions?
If $\mathsf{NP} \subseteq \mathsf{P}/\mathsf{poly}$, then $\mathsf{SAT} \in \mathsf{SIZE}[O(n^k)]$ for some fixed constant $k$. The claimed results should follow by using this circuit to replace the $\mathsf{NP}$ oracle(s) involved in the relevant classes. For example, (2) follows by noting that $\mathsf{ZPP}^{\mathsf{NP}} = \mathsf{ZPP}^{\mathsf{SAT}}$ and then replacing every call to the $\mathsf{SAT}$ oracle with the assumed small circuit for $\mathsf{SAT}$.