Minimum cost flow and Ford-Fulkerson
No.  Ford-Fulkerson cannot be used to solve arbitrary linear programming instances.  It can only solve instances that are in the form of &quot;max flow in this flow graph&quot;.  The dual doesn't have that form.
The dual is to find the minimum cut.  A standard way to find the minimum cut is by finding the max flow, and then using the max-flow min-cut theorem.