I wonder if the probability is dependent only on $r_i$, or also dependent on
the placement of $R$ within $S$?
In these two examples,
it takes an average of $2.70$ steps to reach slicing $R$ on the left, but
$3.16$ steps on the right.
I realize I'm ignoring your condition that $s_i \gg r_i$.
If you are interested in tangles in the sense of Robertson and Seymour, this is just to provide some perspective on it. I am working on this for my Ph.D. project and I thought maybe it is considered helpful if I share this high-level, intuitive perspective here (it is not a detailed definition):
The best and shortest description, I think, is given in the ...
Seymour and Robertson have indeed said that, and in fact they wrote that in their 2003 article in which they published the graph structure theorem.
Here is the quote from Robertson and Seymour „Graph Minors. XVI. Excluding a non-planar graph“ (Journal of Combinatorial Theory, Series B, Vol. 89, Issue 1, Sept. 2003, pages 43–76, doi:10.1016/S0095-8956(03)...
For every $k\ge 2$ there is a non-rectangular shape allowing a uniform folding for all $n$ that are multiplies of $k$:
The reason is that you can uniformly fold it into a rectangle with $k$ layers.
If you are looking for convex shapes, then $k=2$ above is convex.
Here is a non-rectangular convex shape admitting a uniform folding with three layers.
If you ...
It might be hard to find literature on the specific families of packing problems you mentioned. However, they are special cases of a more general pattern of asking for the optimal packing of $N$ congruent copies of an object inside another object. You can find many papers tackling this type of problem, see what kind of computational methods they use, and try ...