Network flow - minimum flow through edge
Yes, it is possible to solve such problems efficiently.  The minimum-cost circulation problem is a generalization of the standard network flow problem, which allows you to set both lower bounds and upper bounds on the flow through each edge.  (You can set all costs equal to 1.)  There are polynomial-time algorithms to solve instances of the minimum-cost circulation problem.

I don't know if there's a simple way to do with a standard algorithm for network flow.