Also, "Kumo's war" and the Third Shinobi World War are the same thing. Even though the major fighting stopped by the Nine Tails Attack, the villages were still technically in a state of war for awhile afterward (three years, given that the signing happens on Hinata's third birthday), and the peace treaty Konoha and Kumo signed in the lead-up to Hinata's attempted kidnapping and Hizashi's death was a peace treaty to end their conflict that began during the Third Shinobi World War.

No, they're different. Remember Neji vs Naruto and Neji vs Hinata? The parade in which Hinata was nearly kidnapped by Kumo on the night after her third birthday? The dispute that ended in Neji's dad's death? The Third World War ended because Iwa and Konoha had a cease fire agreement, iirc. Kumo was still at war with Konoha, though, until after the Hyuga Affair on/following Hinata's third birthday. Kiri wasn't on good terms with them either, given Shisui and Ao fought as enemies.

Furthermore, the Third stepped down after the war and the Fourth was made Hokage. That's a line separating the world wars, when the Fourth takes office.

Long story short, just because there isn't a world war going on (which would've drawn in Iwa and Konoha's allies), that doesn't mean there aren't smaller wars between countries going on. Its a ninja world and war brings in money.