Reducing a minimum cost edge-cover problem to minimum cost weighted bipartie perfect matching
If all edge weights are non-negative, then the minimum weight set of edges that covers all the nodes automatically has the property that it has no three-edge paths, because the middle edge of any such path would be redundant. If we assign each vertex to an edge that covers it, some edges will cover both of their endpoints (forming a matching $M$) and others will cover only one of their endpoints (and must be the minimum weight edge adjacent to the covered endpoint). If we let $c_v$ be the cost of the minimum weight edge incident to vertex $v$, and $w_e$ be the weight of $e$, then the cost of a solution is $\sum_{v\in G} c_v + \sum_{(u,v)\in M} (w_{(u,v)}-c_u-c_v)$. The first sum doesn't depend on the choice of the cover, so the problem becomes one of finding a matching that maximizes the total weight, for edge weights $c_u+c_v-w_{(u,v)}$. If you really want this to be a minimum weight perfect matching problem, then instead use weights $w_{(u,v)}-c_u-c_v$ and add enough dummy edges with weight zero to guarantee that any matching with the real edges can be extended to a perfect matching by adding dummy edges.

If the input graph can have negative edge weights, then the three-edge-path constraint becomes meaningful. In this case it's not obvious to me that there is a polynomial time solution.