Answer by bof for For any finite simple graph $G$, can we always find an intersection graph of some set such that it is isomorphic to $G$?

I don't understand your solution. Here's how I'd do it. Let $G=(V,E)$ be a simple graph, finite or infinite. Let $S=V\cup E.$ (Of course, if $G$ is finite, then $S$ is finite.) For each vertex $v\in V$ let $f(v)=\{v\}\cup\{e\in E:e\text{ is incident with }v\}\in P(S)$ and let $S_G=\{f(v):v\in V\}\subseteq P(S).$ Then $f:V\to S_G$ is bijective and, for distinct vertices $v_1,v_2\in V,$ we have $v_1$ adjacent to $v_2$ if and only $f(v_1)\cap f(v_2)\ne\emptyset.$ That is, $f$ is an isomorphism from the graph $G$ to the intersection graph of $S_G.$