Indeed. But the work of Genauer and Stoltzfus (arxiv.org/abs/math/0511003) has at least a small amount of cluster flavour. And one can also note that the cluster theory for surfaces now involves skein relations.

This has been asked a long time ago (2005), see page 13 of aimath.org/WWN/braidgroups/braidgroups.pdf

Can you please give the exact reference to "a certain paper" ?

The articles by Sefi Ladkani may be relevant, for instance arxiv.org/pdf/0705.0946v2.pdf

Have you looked at what happens when one superposes two such things ? I have seen this idea in some articles on dimer tilings, where one gets loops by gluing the two layers. Here you should also gets lines joining the monomers. There is a similar idea (looking at pairs of objects) in the story of aztec tiling, if I remember well.

A flat and torsion-free connection on the tangent bundle is also called an affine structure. The Levi-Civita connection of a Riemannian metric is always torsion-free.