Equivalent spring-constant for infinite square grid of springs

I'll answer only the third one (for now at least); the movement with limit to small vertical oscillations will be governed by the drum equation:

$\ddot{s}(x,y)=c^2 \nabla^2 s(x,y)$

where $s(x,y)$ is a vertical displacement in point $(x,y)$ and $c$ is the weave speed; using dimensional analysis I would say that $c\sim\sqrt{\frac{k}{\sigma}}$, where $\sigma$ is the mass density. Of course everything is getting much more complex with larger amplitudes.