Consider a simply connected rectangular region $D$ with a uniform rectangular mesh. Then, after the conformal mapping $f$ ,we can obtain a single connected region $G$. At this time,
Where are the maximum and Minimum "grid" in $G$? Are they at the boundary of region $G$ for any conformal mapping $f$?
Do you mean by " 'grid' in $G$ " the image of a mesh square in $D$? If yes you should say so. Your conjecture then is true "in the limit": If a mesh square in $D$ has center $z_0$ and area $h^2$ then its image quadrangle has area $\approx|f'(z_0)|^2 h^2$. Now use that $z\mapsto f'(z)$ is a nonvanishing analytic function in $D$.