Disproving or proving claim that if VCdim is "n" then it is possible that a set of smaller size is not shattered
I agree, the claim as written is incorrect.  If $C^*$ is shattered by $\mathcal{H}$, and $C\subseteq C^*$, then $C$ is also shattered by $\mathcal{H}$; to be possibly over-pedantic:
For each $B\subseteq C$, because then also $B\subseteq C^*$, we have that there is an $H\in\mathcal{H}$ such that $C^*\cap H=B$.  Note that this implies $B\subseteq H$.  Now, $C\subseteq C^*$ implies that $C\cap H\subseteq C^*\cap H$, so we have $C\cap H\subseteq B$.  And having both $B\subseteq C$ and $B\subseteq H$ implies that $B\subseteq C\cap H$.  So $B=C\cap H$.