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
Probably my favorite are the Myriahedral projections[1], basically the Dymaxion map, but with ridiculous number of faces. In his paper he also unfolds a fractal[2].