While the faces are always planar, triangle meshes are actually more problematic than quad meshes for offsetting - with quad meshes there is this specific conical vertex condition you have to meet, but at least if you can optimize for that it does allow freeform shapes, provided the mesh has the right placement of non-valence-4 vertices and directions. For triangular meshes, if you want planar offsets that node out without the faces clipping each other, you can basically only do spheres, planes and extrusions.
Be aware if you are applying the conical quad mesh definition to other surfaces you’ll usually need to add some more goals to keep it smooth and the faces well shaped. With your shape here it worked even without this, because the vertices only needed to move a tiny distance. If you need help with this post again and I’ll point you towards some more examples.
and yes - the planarizing will work also for hexagons (you can use the FaceBoundaries component and connect the discontinuities to a CoPlanar goal). Note there’s no need to optimize for conicality there - For hexagons meeting 3 around a vertex, the offset and noding out is easy, since 3 planes always intersect in a point (except in a few degenerate cases such as when 2 are parallel).
However, when hexagons are planarized they have to distort into some odd shapes on surfaces which are not purely dome-like (synclastic curvature), and in the developable or anticlastic regions you have to have some of these awkward ‘double M’ shaped concave hexagons . Geometrically it isn’t possible for them all to stay convex. You can apply additional goals to keep the edge lengths and angles reasonable, but the concave shapes can’t be avoided.