This an most likely happen after World War 1 with some King of Kurdistan imitating the Arabs down south and wining against them. They'd lose their Turkish part in a brutal war with the Ankara government, a competent regime, like the reconquest of Kars and Smyrna.
The reduced Iraqi-Syrian Kurdistan (maybe a bit smaller or bigger) would have an economy based around Arbil oil after the 20s and be considered the least prestigious of the Middle Eastern Kingdoms because it's not so easy to govern this land (today they're divided too) and the Arab Monarchs have more western view. Big spending will be done on the military, building up a considerable but ragtag and backward force. Hashemites and Turks will both have designs, so finding British or Soviet supplies would be hard. Their best bet is with the French.
Ataturk's regime took peace after initial wars as a direction, and Kurdistan would follow them into armed neutrality in world war 2. After the war, it would turn into another Jordan, an armed, army-loyal monarchy. Independence of Syria and resulting instability will mean Arabs agitating on Kurdistan and total loss of aid. It would be an Israeli ally, though that's complicated by Turkey also being close to Israel. Iran would be hard to buy because of the Kurds in Iran and Kurdish irridentism. USA meanwhile will supply Turkey.
Kurdistan after the Arab nationalist revolutions (if it doesn't have a revolution and become republic or socialist state, which I see no reason for it to - Soviets invading Iran established a Kurdish republic, but not a popular one), would have newfound British support, and normalized relations with Turkey and Iran, to a degree. This will eventually phase out to American support.
Kurdish economy would develop along the lines of the Libyan, with big income inequality despite oil, but the conservatives wouldn't be overthrown because there isn't an Israel problem. Arab-Kurdish wars are a possibility, and Kurdistan wouldn't lose them because their supplies are probably better, and their doctrines from their military culture and years of militarism would be too. I see it becoming either a Turkey-style republic, though not necessarily democratic, or a constitutional monarchy.
IS would develop normally, and I don't think Kurds would intervene in Iraq or Iran. They might be subject to an invasion by Saddam, which might devastate the country in the meantime. If that's the case, a dictator would come to power, and deal with the US and USSR as any other middle eastern strongman.