Complexity of computing a densest minor
Ok, since there's still nothing here in the way of an answer, let me at least make a couple of simple observations:

For graphs of bounded treewidth it should be possible to find a densest minor (or even a minor with specified numbers of edges and vertices) by the usual sort of dynamic program on the tree decomposition, where each state of the dynamic program keeps track of the number of edges and vertices in the part of the minor living in a subtree of the decomposition, the subset of vertices in the bag of the decomposition that participate in the minor, the equivalences between vertices in this subset caused by the minor contractions in the whole graph, and a refinement of this equivalence relation caused by the contractions in the part of the minor living in the subtree.

If so, it would follow that, when the density is below three, it should be possible to find the densest minor in polynomial time (with a constant factor that depends on how close to three the density is). For, the graphs whose densest minor has density $\le 3-\epsilon$ have planar forbidden minors and therefore bounded treewidth.