I was wondering whether or not Sea Kings know about the world: what is going on (when there was the Marineford war for example), who the world rulers are, how the power is balanced in the world, etc. And if they do, then how, when they can't speak to anyone from the surface? Or do they understand humans even though humans can't understand them (honor the exceptions)?
The short answer is: as for now (chapter 967), there is no concrete answer to your question in the manga.
If we look a bit more deeply into the chapters and try to interpret the information given, then it gets obvious that some form of communication occurs between Sea Kings and "human beings" (including life forms not necessarily called humans in the manga such as fish-men).
It is being said that the antic weapon Poseidon gets reborn every (few) 100 years in the form of a mermaid princess and will lead the Sea Kings (chapters 648, 650, 967). This stands on the poneglyphs that are at least 800 years old so around 8 people were able to directly communicate with the Sea Kings (but not necessarily doing so) like princess Shirahoshi did.
It was shown that Gol D. Roger and possibly Luffy can hear their voice too (chapter 648 and 967). Though this did not lead to a conversation, they may be able to converse. Roger was also able to hear other "voices" such as the one of the golden bell in Skypiea/Shandora and Zunesha's in Zou. Oden and later Luffy and Momonosuke were able to feel respectively hear Zunesha's voice too. Mononosuke is so far the only one that was shown to talk directly to Zunesha (chapter 821) so he might be able to talk to Sea Kings too. That's an indirect way the Sea Kings could get informed about the whereabouts of the world.
The Sea Kings know about the history of Poseidon so either a person told them or - if this story actually originates from ancient Sea Kings and not from humans - they must have told someone for this story to be transmitted in the form of poneglyphs among humans.
Finally, Sea Kings go to the surface from time to time, even close to islands (chapters 101 and 319 for example), so they might get information about what happens on land on their own, looking around and bringing these information back to other Sea Kings. From the level of communication they show in chapter 648, one can safely assume that they are smart enough to understand whatever human activity they can observe.