SK is still on war with NK, and starting from 2017, NK is constantly threatening SK by their nuclear weapons. Also, there is no right to other countries that they can tell SK to not produce their own nuclear weapons. Also there is no specific reason why SK isn't producing their own nuclear. AS
The reason South Korea has nuclear weapons is probably because they are still at war against North Korea (yes, SK is still at war with NK). Since NK is not open and about, there might be a small chance that they have nuclear weapons too (at least according to my sources). So yes, it's mostly for self defense probably.
There is no exact reason why S.Korea is not allowed to have nuclear arms. And even if S.Korea builds up nuclear weapons, why does it concern other nations? S.Korea is a petty nation everyone thinks it is. And it isn't going to sit around letting N.Korea keep threatening it. If S.Korea begins to stockpile nuclear weapons, maybe it'll set the table from being to one sided. Giving each Koreas an equal grounds. I bet N.Korea wouldn't be talking crap if S.Korea begin and already mass produced nuclear weapons.
S.Korea already have a strong relation with China, a stalemate one sided relation with N.Korea, and with its new foreign relations, until Japan improves their foreign policy. Japan is basically a simple dust to Korea, so whats the problem? All the resolution are either improving or at least equalizing the standings in the Asian mainland.
If South Korea wants to stockpile its own nuclear weapons, then no other nation should stop them. The United States and other world powers own these weapons en masse, and keeping them out of a secure nation makes no sense. Although reducing the number of missiles available between countries is ideal, such an event is impractical in today's world.
South Korea should not be allowed to have Nuclear Weapons because that would encourage other nations around the world to pursue in the same direction. It would encourage other Asian nations such as Japan, Hong Kong, Phillipines to want to have nuclear weapons as well, which would cause more problems than there currently is.