WWII is in no relationship to this.
The start you can date before though.
British-ness is the reason for the green of the British-ish parts.
EDIT: Correction, I think BJ is right that Japan's green-ness is caused by WW2, but WW2 should otherwise not be important for this.
EDIT2: this point is a bit complicated...
Some association status of an international organization?
Currently no, although maybe at some point there might have been a faint relationship. (and I don't think this would help)
Any hint at the order? I would guess red > yellow > green, gray no data? And you have data for Laos?
I took the map from Wikipedia, so I can't be sure about the data for Laos, but I'd say there's a good chance that it's valid. EDIT: Checked another website, says it's correct for Laos

.
No real order, but categories.
Grey might be no data, but in most cases it probably validly means absent.
South Korea grey, Best Korea red is different.
This is correct, and might be of importance.