For a flow network, is it possible to show that there always exists a maximum flow which would assign integer values to all the edges?
If edges have real valued capacities, there may not exist a maximum flow which assigns an integer value to every edge. For example consider a single edged $s-t$ network with $c(s,t)=0.1$ where $c$ is capacity function. However if edges have integer valued capacity, there exists a maximum flow which assigns integer value to every edge because in each iteration of ford-fulkerson algorithm, each edge in residual graph is assigned an integer value. So the result is an integer.