The first equation comes from a calculation of the potential V as a function of position z, and then taking the derivative of epsilon_0 V(z) with respect to z and setting z=0. Here's a description of that:

https://en.wikipedia.org/wiki/Method_of_image_charges
As for the second equation - you're right that the equation for force has a factor of 4, but here the distance between the two particles is 2a, so that distance squared becomes 4 a^2. Hence the equation ends up with a factor of 16.