Huge post ahead, sorry:
To be honest, if you aren't korean or japanese you shouldn't care at all about Dokdo. That's kind of the point when they talk about it here. It's not that it's korean or not but the thing that they don't give a fuck about it but they're having it constantly shoved through their mouths because, for some koreans, international support give legitimacy to their claims (and not having been on the island for the last 60 years, surprisingly). Is fair enough for them, if they're defending one position, to not present the other side; that would be stupid, like incriminating yourself to a crime. Just say that you don't know about it and go on with your life, nobody is going to force you to have an opinion more than the marketing campaign they have.
About the anti-japanese sentiment, it's something you have to deal with because it isn't going to disappear in a couple of years. Korea was occupied and humiliated during 35 years, destroying a great part of their culture assets (including trying to reduce the use of the same language you're learning now) and traditions while almost using them as slave labour and defeating every attempt their liberation movement did to be free. This is a real complicated moment of their recent story and you can't really learn a language without know the story behind it. Just 5 years after being liberated the country was broken in half and destroyed by the war so until today they have never been the same. Learning the language (to talk to native koreans) without seeing this is like learning the rules of basketball to never play it, completely pointless.
With a huge economic growth but going through 3 different dictatorships and the fear of an imminent new war (today not a single korean is being paranoic about it but they still have forced conscription and a completely overblown military budget and development) i can't really blame them for being a little bit romantic and blaming the japanese for beginning this chain. The japanese didn't help with their constant changes of opinion between begging perdon and calling the comfort women prostitutes, and blaming them for the poor class community of zainichi koreans and for not getting over what happened so long ago.
I don't why it makes your image of korea worse, because that's the real image of korea.
TL;DR: The point is that no one is forcing you about it, it's a important inseparable complex part of recent korean culture and you should learn to study its implications if you really want to learn their culture.