Polynomially Equivalent Pairs of Minimization-Maximization Problems in Weighted Graphs
For item (b), the reduction does not work. To understand why, imagine a graph with all edges having the same weight. Then, the reduction has no effect: $G = G'$. Finding the shortest path from $u$ to $v$ consists in finding the path with fewer intermediate nodes, while finding the longest path, consists in finding the path with the highest number of intermediate nodes. The problems are not polynomially equivalent because the shortest path is in $P$ while longest path is in $NP$ (see here).
For item (c), the optimization problem for MAX CUT is NP-hard while for MIN CUT is polynomial. However, when you consider the version of these problems that allows both positive and negative weights they can be trivially converted to each other by flipping the sign in all weights.