Is it true that if I multiply all the edge capacities in an s-t flow problem by a positive constant k > 0, then the maximum flow increases by the same factor of k.
Yes it is
the sum of the capacities of the arcs on the minimal cut equals the maximum flow is a famous theorem of network theory called the max flow min cut theorem
Let minimal cuts of arcs in s-t flow are ...
This verifies a statement regarding edge capacities and maximum flow.