**Question 1**: No, $(R^{A_7} \subset R^{S_4})$ is a counterexample.  
Thanks to the subgroups lattice of $A_7$ (see [here][1]), 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 $(R^{S_4} \subset R^{A_7})$ 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?][3]

**Remark**: It could be relevant to add the assumptions *abelian* and *Dedekind* for being a cyclic subfactor.  
(see  [Abelian subfactors, a relevant concept?][2])   




  [1]: http://homepages.ulb.ac.be/~tconnor/atlaslat/alt7.pdf
  [2]: http://mathoverflow.net/questions/156374/abelian-subfactors-a-relevant-concept
  [3]: http://mathoverflow.net/questions/156311/jordan-holder-theorem-for-subfactors