[m1el]

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

[lukewtollefson]

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].

[1]: http://www.win.tue.nl/~vanwijk/myriahedral/

[2]: http://www.win.tue.nl/~vanwijk/myriahedral/CAJ103.pdf

[mitchtbaum]

I like Cahill-Keyes: http://www.cartotalk.com/index.php?showtopic=9406 (d3.js animation included).

[mjmdavis]

I think Healpix is included in the tool at the bottom. Also check out the last few in the list which are similar to Dymaxion.