[dvgrn]

-- It's interesting to look at the different symmetries on Catagolue, where the idea is to generate a lot of random soups with different symmetry types, and then just run those patterns and see what comes out. The D8_1 and D8_4 symmetries are 8-way symmetric, which is as high as you're going to be able to get on a square grid.

  https://catagolue.hatsya.com/census/b3s23/D8_1
  https://catagolue.hatsya.com/census/b3s23/D8_4

Scroll down to the bottom of those pages and click on, especially, some of the higher-period "xq{N}" categories. These are objects that showed up "naturally", evolving from random soups.

-- There have definitely been a number of people over the years exploring various outer-totalistic rules on a hex grid, and (to a lesser extent) isotropic non-totalistic rules: see

  https://conwaylife.com/wiki/Hexagonal_neighbourhood

-- The smallest period at which gliders can follow one another is period 14. We don't have a true period-14 gun yet, though. The closest we have is a "pseudo-period" gun -- actually period 28, but it generates two gliders per period, so you end up with a period-14 stream:

  https://catagolue.hatsya.com/object/gun_14/b3s23