As for the military issue, since someone asked for a recap, basically Son is stuck in a sh!tty confluence of factors working against him. He was born too late to exploit a loophole for middle school dropouts which Lee Chung-Yong (formerly at Bolton, now at Crystal Palace) used to gain exemption. Now that loophole is apparently reopening but is only open to people born after 1996. He missed exemption with the 2012 Olympic squad (sorry for knocking out Team GB) that won Bronze and then missed it again in 2014 when Korea won Gold in the Asian Games. That one hurt especially, since not only was it one of his last chances to gain exemption, the Korean FA told Leverkusen that Son only had to show up for the Final. Basically they were willing to make a farce of the competition in order to get Son his exemption. Bayer refused to release him for that single game, his future be damned. Now he only has the 2016 Olympics and the 2018 Asian Games, unless Korea makes some serious noise in the 2018 World Cup. Given how crap we were in Brazil 2014, I wouldn't be banking on that option..
As for deferring service based on residency or fabricating a medical loophole, there is a long string of celebrities/pop stars/etc. who have destroyed their careers by attempting such things. Son would effectively become persona non grata. The question remains if Son can become a big enough star to force the issue and sway public opinion if the time comes for him to report for duty and he hasn't gained exemption. It would pretty much take an Act of Congress (sorry, don't know what the Korean equivalent term is) to get him an exemption by fiat. Even then there would be very vocal opposition, regardless of how big a star Son might be.