My question is about $m \times n$ binary matrices (aka $\{0,1\}$-matrices), whose rows all sum to the same value, and whose columns all sum to the same value (but these two values may be different).

The first question is simply: is there a standard name for such matrices? They correspond to the biadjacency matrices of so-called "biregular bipartite graphs", but this terminology doesn't appear to be commonly used.

Second, are there any "interesting" constructions of families of such matrices, in particular that are connected to other combinatorial objects?

Two simple examples of constructions of these matrices are the $\binom{n}{k} \times n$ matrix whose rows consist of every $n$-bit string with Hamming weight $k$; and the $2^n \times 2^n$ Sylvester-Hadamard matrices with the first row and column removed.

I did find a paper by Brualdi titled "Matrices of Zeros and Ones with Fixed Row and Column Sum Vectors", but this seems to be more concerned with the question of existence of these matrices, rather than constructing them.