Here is a proof of Conjecture 1.   

**Proof.**  We proceed by induction on $|V(G)|$.  We may clearly assume $|V(G)| \geq 4$.  Since $\mathcal{C}(H) = \mathcal{C}(H)^{{\perp}{\perp}}$, for all induced subgraphs $H$ of $G$, we may assume (by induction) that every proper induced subgraph of $G$ is a cograph.  Thus, $G$ is itself a cograph unless $G=P_4$.  Let $V(G)=\{1,2,3,4\}$ and $E(G)=\{12,23,34\}$.  Thus, $\mathcal{C}(G)=E(G)$ and  so $\mathcal{C}(G)^{\perp}=\{13,24\}$.  It follows that $\mathcal{C}(G)^{{\perp}{\perp}}=\{12,23,34,14\} \neq \mathcal{C}(G)$, which is a contradiction.