There seem to be some convention issues here which would be alleviated by including the relevant matrices for n=3 and n=4 in the question. (I did try to download the suggested file, but for whatever reason was not able to gunzip it.) If the 1st row is really supposed to represent the situation after 1-1=0 partitionings, then it would be a vector of length $n$, with a 1 in the final position, contrary to the assertion that the matrix is $n-1\times n-1$. Also, it seems to me that the resulting matrices are not upper triangular since they are above the antidiagonal not the main diagonal.

I don't think your formula for the distance between the identity and the longest permutation is correct. $w_n$ shouldn't appear at all, and, unless I'm misunderstanding, $w_1$ would appear twice, once for position 1, and once for position $n$.

Yes, sorry, in the contexts I work in, irreducibility of the root system is a pretty harmless assumption.

Oops, actually $2^{r-1}$ simply is the number of Coxeter elements. A choice of Coxeter element requires precisely that you specify, for each pair of non-commuting simple reflections, which one comes first. (The order of commuting reflections doesn't matter since they commute.) There are $r-1$ such pairs, and since the edges of the Dynkin diagram form a tree, any choice of orders is obviously realizable. Eg for A_3, the four choices are $s_1s_2s_3$, $s_3s_2s_1$, $s_2s_1s_3=s_2s_3s_1$, $s_1s_3s_2=s_3s_1s_2$. In the last of these, for example, we chose $s_2$ to come after $s_1$ and $s_3$.

This question could be improved by including a definition of fair division of a necklace. (While I'm asking, a link to the well-known statement that at most $d(p-1)$ cuts are required would also be nice.)

The number of elements in $W^P$ is the number of elements in $W$ divided by the number of elements in $W_P$.

The number of Coxeter elements is bounded above by $2^{r-1}$. Could this also be a lower bound for the multiplicities? It seems like $2^{r-1}$ is a lower bound for the Kostant partition function of $\lambda-\mu$ once we know that all simple roots appear in its simple root expansion, but on the face of it that isn't quite enough.

This question will likely attract more attention if it is clearer what you are really asking. "But what about is the base ring of of" comes at a key point and doesn't make grammatical sense, which isn't helping.

Yes. Note that what some people call (A,X), others would call (X,Y).