Packing - some more thoughts

1. Which convex hexagon is worst for packing (ie leaves largest fraction of plane open)?

2. it has been conjectured that the regular heptagon is the worst convex region for packing. Is it known if regularity is significant? Indeed, for hexagons, regularity gives a perfect pack.

So, one can ask: among all convex heptagons, is the regular one the worst for packing?

Further, is there any reason to believe that the worst packing region has SOME rotational symmetry?

And what is the connection, if any, between a body being bad for packing and bad for COVERING?