The cyclic subfactors theory defines a too large class$^{\star}$ of subfactors for being an *good* quantum arithmetic (i.e.  quantum generalization of the **natural numbers** theory).   
Nowadays, my better attempt for this is the **natural subfactors** theory (see  the *optional part* [here][1]).    

$^{\star}$Up to equivalence, more than $70$% of the inclusions of groups of index $≤30$ are distributive lattice,  
 so we can interpolate that in the general subfactors theory, the **cyclic subfactors are majority** !

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**Question 1**: No, $(R^{A_7} \subset R^{S_4})$ is a counterexample.  
Thanks to the subgroups lattice of $A_7$ (see [here][2]), 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.    


See the questions: [Jordan-Hölder theorem for subfactors?][3]  and [Abelian subfactors, a relevant concept?][4]

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**Question 3**: No, the maximal subfactor $(R^{\mathbb{Z}_3 \rtimes \mathbb{Z}_2} \subset R^{\mathbb{Z}_2})$ gives counter-examples.   
It is depth $4$ and it admits **several** compositions with itself which are also   cyclic subfactors of depth $4$,    
for example $(R^{(\mathbb{Z}_3 \rtimes \mathbb{Z}_2) \times (\mathbb{Z}_3 \rtimes \mathbb{Z}_2)} \subset R^{\mathbb{Z}_2 \times \mathbb{Z}_2})$ and $(R^{\mathbb{Z}_9 \rtimes \mathbb{Z}_2} \subset R^{\mathbb{Z}_2})$.  
 Their intermediate subfactors lattices are as $\mathcal{L}_6$ and $\mathcal{L}_4$, so they are not isomorphic.   
($\mathcal{L}_n$ is the lattice of divisors of $n$).


  [1]: http://mathoverflow.net/questions/160577/are-the-homogeneous-single-chain-subfactors-dedekind
  [2]: http://homepages.ulb.ac.be/~tconnor/atlaslat/alt7.pdf
  [3]: http://mathoverflow.net/questions/156311/jordan-holder-theorem-for-subfactors
  [4]: http://mathoverflow.net/questions/156374/abelian-subfactors-a-relevant-concept