If you fix $m,n$ and the row and column values, and then consider the matrices as points in $\mathbb{R}^{mn}$, their convex hull forms a convex polytope known as a transportation polytope. In the particular case of permutation matrices one obtains the famous Birkhoff polytope for instance.
So I would advise you to look up references about those polytopes, since your matrices arise naturally as their vertices; hopefully you can find the names and particular constructions you asked for.
(Well this this should probably be a comment, since I do not really answer any of your questions; unfortunately I do not have enough magic points to write one)
Edit: As pointed out in a comment, what I wrote above is simply not true. In fact, the $\lbrace 0,1\rbrace$-matrices considered are more precisely the lattice points of the intersection of a transportation polytope with the hypercube $[0,1]^{mn}$; this forms another polytope which may actually be interesting in its own right.
