Weighted cycles in weighted line graphs
For the second problem, this question is no harder than finding a cycle of a specific weight k in a general weighted graph in the following sense: consider the line graph of G, that is, a graph L(G) with V(L(G)) = E(G), and two nodes u,v in L(G) are adjacent if the corresponding edges in G are incident to each other. By assigning weights to the edges of L(G) according to your definition, it reduces to the normal weighted cycle problem.

Thus to the first problem, in general if we do not have any constrain on k, we can assign 1/|V(G)| to each edge in an unweighted graph G, and reduced the Hamiltonian cycle problem to the weighed cycle problem by asking k=1, thus it is NP-hard. If you are asking non-negative integral weights, then the algorithm mentioned in the MO post again works since we can subdivide the edges into an unweighted graph. But be careful since the size of the subdivided graph will depends on the total weight on the original graph.