That is the beauty of it. Everyone assumes Gary Oldman's character will start the war, because "Rise" established in this new franchise the apes are the more "noble" and just species (though that is in the original too if you actually seriously consider Taylor's actions and the implicit cultural horror of the first movie's ending).
But as with any "stalemate" situation where two cultures or different groups of people try to share the same space, there will be bad actors and war hawks on either side. The movie is not "apes are good, humans are evil" like the first one kind of simplistically implied. As with almost any war or conflict, there are actors on both sides that push for violence and conflict based on prejudices, such as Koba, who proves far more militaristic than Oldman's character.
As for the immunity, some people genetically could have an immunity to it. This is a virus that apparently wiped out 99% of the world's population or something absurd like that. Even the catastrophic Bubonic Plague of the early 14th century only wiped out roughly 1/4 to 1/3 of Europe (though some communities lost up to half their populations). That is still end of the world type horror, but there were people who survived that at much higher numbers than this film's rather grim scenario.