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What you are studying is the lattice of normal subgroups of the free group $F(X)$. The normal subgroup lattices of groups have been studied a lot. For example, this lattice is complete and modular. See the references here.

Update. About your newer, more concrete, questions. Even if $R$ and $S$ are finite, the normal subgroup $\bar R\cap \bar S$ may not be finitely generated. If the membership problem for $\bar R$ and $\bar S$ is decidable, then in principle one can decide the membership problem for the intersection, so you will find a generating set of the intersection - the whole intersection. But I am sure that the problem whether the intersection is finitely generated is undecidable (although I did not think about a proof).

The question whether $\bar R\bar S=F_k$ is the triviality problem which is undecidable (by a theorem of Adian-Rabin) even when $R$ is empty. For example, take any presentation of the trivial group, call a subset of it $R$ and the complement $S$. You get the equality $\bar R\bar S=F_k$. Another way: take a presentation of a finite group of exponent $n$, say, $S_3$ has exponent $6$ and presentation $\langle a,b \mid a^2=b^3=1, aba=b^{-1}\rangle$, and add relations that "kill" the generators, say $a^5=1, b^7=1$. The first set of relations is $R$, the second is $S$.

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What you are studying is the lattice of normal subgroups of the free group $F(X)$. The normal subgroup lattices of groups have been studied a lot. For example, this lattice is complete and modular. See the references here.

Update. About your newer, more concrete, questions. Even if $R$ and $S$ are finite, the normal subgroup $\bar R\cap \bar S$ may not be finitely generated. If the membership problem for $\bar R$ and $\bar S$ is decidable, then in principle one can decide the membership problem for the intersection, so you will find a generating set of the intersection - the whole intersection. But I am sure that the problem whether the intersection is finitely generated is undecidable (although I did not think about a proof).

The question whether $\bar R\bar S=F_k$ is the triviality problem which is undecidable (by a theorem of Adian-Rabin) even when $R$ is empty.

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What you are studying is the lattice of normal subgroups of the free group $F(X)$. The normal subgroup lattices of groups have been studied a lot. For example, this lattice is complete and modular. See the references here.