Minimization of Essential matrix
I guess this is a straightforward non-linear optimization problem (to be solved with Newton variations, such as Trust-Region methods), where you don't even need to compute the Jacobian analytically. It appears to me that the optimization problem is written over $K_i$, and thus is the input to the cost function. To compute the cost, at each call to this function, you basically compute the singular values of $K_i$ and compute the cost according to equation $4$ (in the paper). As your input parameter is $K$, the derivatives are computed over the elements of $K$. That makes your optimization $4$ dimensional (or $5$ if you consider the skew) per camera. The derivatives are computed automatically, and you don't need to care about that. If you are using MATLAB,lsqnonlin would work for you.

The weight computation is explained in a detailed fashion in the paper, so I skip this part.

Having checked the paper once again, I have noticed that the authors are actually using the numerical differentiation scheme I have mentioned. If you would like to understand more deeply how to differentiate an SVD, you might like to check out this one or this.