Regarding **Question 3**, here is a proof that $f(n)=30$ for all $n \geq 40$.  Let $H$ be a $2$-connected planar graph with minimum degree $5$.  Let $G$ be obtained from $H$ by adding a new vertex inside each face of $H$, and making it adjacent to all vertices of the face.  Let $V(G)=X \cup Y$, where $X=V(H)$, and $Y$ are the newly added vertices.  Since $H$ has minimum degree $5$, $\deg_G(x) \geq 10$ for all $x \in X$. Moreover, $\deg_G(y) \geq 3$ for all $y \in Y$ and no two vertices of $Y$ are adjacent.  Thus, $\deg(u)\deg(w) \geq 30$ for all $uw \in E(G)$.  Note that $G$ is a planar graph with $|V(H)|+|F(H)|$ vertices, where $F(H)$ is the number of faces of $H$. By Euler's formula, $|V(G)|=|E(H)|+2$.  Since there is a $2$-connected planar graph with minimum degree $5$ and $m$ edges for all $m \geq 38$, we are done.