Someone else might be able to provide you with more information but I believe kurdish forces were involved in the Anglo-Iraqi war in 1941, I have struggled to determine from a reliable source about any involvement of Kurds in the Lebanon Syrian campaign however. Meanwhile in Iran there was a parallel conflict to the second world war with a Kurdish uprising called the Hama Rashid revolt that occurred during the Anglo-Soviet invasion of Iran.

So the Anglo-Iraq war first. In 1941 there was a coup instigated by four major members of the Iraqi military known as the golden square designed to produce a Iraqi-Arab nationalist government aligned with the Axis powers. Commonwealth forces fought against the Golden Square forces including a unit known as the Iraqi Levies who were an infantry unit made up of an assortment of ethnic minorities native to Iraq including Kurds who fought under the command of the RAF. One of the major engagements they fought in was the defence of the RAF base in Iraq, Habbaniya when it was besieged. As an RAF unit they continued to primarily be used to guard RAF airfields in both Iraq and Cyprus over the course of the war.

This facilitated further operations in the region. The RAF taking part in the Syria–Lebanon campaign when the allies drove the fascist aligned Vichy French Forces with German air support from the French colonies.

The base continued to be an important part of the supply from Egypt during the Anglo-Soviet invasion of Iran when Commonwealth and Soviet forces ousted Reza Shah who was considered to by an axis sympathiser in order to use the Iranian oil fields for the war effort and allow allied forces a route to supply the Soviets from the south in their efforts on the eastern front.

Concerning Hama Rashid revolt i'd be interested to see what anyone else has to say on the subject as i don't feel qualified to speak in much detail about it.

Missed a "do" after the actually. Dammit.