[sharpener]

An untested idea/suggestion...

In the first Open Problem, maybe the generator tweak for random Hamiltonian paths is that two grids are needed: one for the eventual H-path; and one of points that are the centroids of the first which is used to draw a random tree starting from some point, subject to some rules about distances to unconnected nearest neighbours being acceptable (hunch e.g. of length >= sqrt(2)). Then draw around that tree on the first grid to get the H-path. (Grid graph duality use.)

I'd guess this can be generalised to 3D by drawing the H-path on each 2D plane slice, and some simple rules about choosing the degree of interconnection between those planes.

It seems like all the tools for tuning the shape of the initial random trees already exist in MJ, based on the examples shown.