[Big-Daddy]

What is so special about having a carbon ring of 6 carbons, that graphite and graphene both exist as "polymers" or at least long arrays of 6-carbon hexagonal rings, as opposed to 5-carbon pentagonal rings, 8-carbon octagonal rings, etc., whereas to the best of my knowledge these latter forms are never found in nature (or are at least vastly less common)?

[Corribus]

Ring strain (or lack thereof) is a big part of it.  A regular hexagon has angles of 120°, which just so happens to be the same as that for the bonds around an sp² hybridized carbon.

There's also the matter of periodicity.  There's a reason strategic board games use hexagons and not pentagons.

[Enthalpy]

For alkanes, C₅ and C₆ rings are about as stable, since the tetrahedral bonds fall between both angles, and C5 rings are common in liquids and gases - but don't form regular 2D crystals.

Now, many hydrocarbons come from fossile deposits where the C/H ratio makes unsaturated compounds, of which aromatic ones are much more stable, for which C₆ rings are good candidates. But again, in liquids, you find plenty C₅ compounds, including some with several fused rings.