Question 1: No, $(R^{A_7} \subset R^{S_4})$ is a counterexample.
Thanks to the subgroups lattice of $A_7$ (see here), there is a subgroup of $A_7$ called $2^2:S_3$ isomorphic to $S_4$, such that $(S_4 \subset A_7)$ has exactly two non-trivial intermediate subgroups: $K=L_2(7)$ and $L=A_4:S_3$, so that $(R^{A_7} \subset R^{S_4})$ is cyclic.
Now if we consider the two maximal chains $(S_4 \subset K \subset A_7)$ and $(S_4 \subset L \subset A_7)$, then the intermediate indices are $(7,15)$ and $(3,35)$, so these chains can't be equivalent.
A more relevant related open question is: Jordan-Hölder theorem for subfactors?
Remark: It could be relevant to add the assumptions abelian and Dedekind for being a cyclic subfactor.
(see Abelian subfactors, a relevant concept?)