Even in the special case $K = \mathbb R$, $n=m$ and $c_{\rm row} = c_{\rm column}$, this is not an easy problem!
A good place to start looking would likely be the following paper.
On the minimum rank of regular classes of matrices of zeros and ones
by Brualdi, Manber and Ross
http://www.sciencedirect.com/science/article/pii/0097316586901135
You may also want to see Section 3.10 of Brualdi's 2006 "Combinatorial Matrix Classes" book. I suspect you will find it contains just about all of what is known.
Below find six suggestions.
Case of general ZLS matrices.
Examples of properties are:
- All cofactors of a square ZLS are equal.
- Each square ZLS matrix of dimension $n$ has eigenvector $1^{n\times 1}$ with eigenvalue $0$.
- If an $n\times n$ ZLS $A$ is moreover symmetric, and if $\mathrm{Sp}(\cdot)$ denotes the spectrum of a matrix, then each cofactor of $A$ equals $\frac{1}{n}\prod_{\lambda\in\mathrm{Sp(A)\setminus\{0\}}}\lambda$.
- Rectangular ZLS matrices over a field $K$ form a $K$-vector space in the obvious way, and according to this thread its dimension is $(m-1)(n-1)$, and if $m=n$, then according to that thread its orthogonal complement w.r.t. the Frobenius inner product consists solely of sums of two rank-one matrices, more precisely, this orthogonal complement consists solely of matrices of the form $v\cdot 1^{1\times n} + 1^{n\times 1}\cdot w^{\text{t}}$.
- Some relevant Lie theoretic consideration can be found in
Andreas Boukas, Philip Feinsilver, Anargyros Felouris, On the Lie structure of zero sum and related matrices. Random Oper. Stoch. Equ. 23, No. 4, 209-218 (2015)
Case of matrices which are differences of two doubly-stochastic matrices
- If $A$ is the sum of two symmetric stochastic matrices, you may find it profitable to look in the direction of Horn's conjecture. Starting points could be R Bhatia: Algebraic Geometry Solves an Old Matrix Theorem. Resonance. December 1999, or this MO thread.
Best Answer
If you are looking for a simple formula, you are out of luck except for small $n$ or $T$. As Sam mentions in a comment, these are the integer points in a dilated Birkhoff polytope. Since the vertices of the polytope are lattice points, the number of matrices is a polynomial of degree $(n-1)^2$ in $T$ for any fixed $n$ (the Ehrhart polynomial) but the polynomial is only known up to $n=9$.
This paper and this paper have some stuff about asymptotics. Searching for these buzzwords will get you more interesting things, like this paper.