Characterizing the set of problems solvable via network flow
It depends on how you define "solvable via network flows". If you allow the preprocessing step to take arbitrarily long, then anything is solvable using network flows by simply solving it an then outputting a trivial network flow that has the desired answer.

Under log-space reductions, network flow is $P$-complete so any problem in $P$ can be solved "efficiently" using a network flow.

Another interesting aspect of flows is that if you start with integer capacities, then the solution will be integer as well. Network Flows (when viewed as Linear Programs) are actually part of a larger class of problems called totally unimodular. The optimal solutions to these programs are always integer.