It would have been better if they looked for the densest circle with an area of 1 square km. Although a square shaped area has its tangible benefits - you "feel" population density more when it’s on your own street obviously, and since square areas provide more information along a few axes (the ones containing the corners), you can get an idea of the population density in an area by placing a square’s corners on the street. A square is probably not the best shape for this, though.
Just think how insanely unlikely it would be for this not to be the case.
One could also increase the density by allowing non square shapes or squares not fixed on the grid and so on. A more general approach would always lead to slightly higher scores.
Perhaps something akin to a Hilbert curve that snakes around the whole country encompassing every individual person’s footprints but otherwise covers almost-zero area.
Yes, a grid system like H3 https://www.uber.com/en-NL/blog/h3/ is is closer to a circle in shape. One of the reasons for Uber to use H3 is bacause density maps look more natural in hexagonal shapes. It would be interesting to see the answer for H3 hexes.
No sorry, just geospatial (horizontal position on the surface of the Earth).
It should be theoretically possible to re-develop it for other spaces (it's "just" a hexagon tiling), but the H3 team put a lot of work into the details and you'd be taking on a big effort redoing that for a different space.
