There's a few map projections I really like that don't have most of these problems: HELAPIx[0] and Dymaxion map[1].

HELAPIx is designed to have pixels of the same area, and Dymaxion is designed to have the least distortion.

[0]: http://healpix.jpl.nasa.gov/

[1]: https://en.wikipedia.org/wiki/Dymaxion_map