You are interested in estimating sizes and constructing independent sets in graphs obtained by merging several classes of the Hamming association scheme $H(n,2)$. That is to say, you consider graphs $H_0$, $H_1$,... $H_n$ on the vertex set $B^n$ of binary $n$-sequences, two vertices of $H_j$ adjacent if the Hamming distance between the corresponding sequences is $j$.
In fact you have an edge partition of the complete graph (with loops) on $B^n$, satisfying an extra property that the linear span of the adjacency matrices of the corresponding graphs is closed under multiplication. This basically is a definition of association schemes. What's nice about that that you can use a lot of algebra/algebraic graph theory on them, to obtain various nontrivial bounds.

What you referred to as "a fundamental problem" then becomes a problem about independent sets in the graph $\Gamma_k$ with the vertex set $B^n$ and edges from $H_1$,..., $H_{k-1}$. You can form other graphs from $H_1$,... $H_n$, and/or take other association schemes.