A second way is to measure the distance traveled by a plane from Mexico to Japan, heading WEST. If the distance is greater than the distance EAST from Mexico to Japan, then it supports the flat earth.
No, it doesn't. In either case, the proportion would be the same.
You are right, the distance from Mexico to Japan does look the same on a flat earth, whether the route goes East or West. On the flat earth map I noticed that the shortest path (starting at Mexico) between the countries is to fly northwest right past the North Pole. But (excuse my ignorance) I don't know if that's possible with real aircraft because I'm not an aviator (or an air traffic guy).
About the South Pole, I found this on Wikipedia:
http://en.wikipedia.org/wiki/Polar_routeLook at the third picture to the right. How is that route possible on a flat earth?
Also I find this comment from Wikipedia interesting: "As it happens, airlines don't fly nonstop between many city-pairs having a great circle route over Antarctica. Direct flights between South Africa and New Zealand would overfly Antarctica, but no airline has scheduled such flights."