I am specifically interested in knowing more about their revolts in the 1920s and 1960s. Who stopped them from becoming a country and why?
Did/do they have a large amount of oil in their defined borders?
Is there anything specific besides land that Iraq, Iran, Syria, and Turkey stand to lose by a separate Kurdish state?
On mobile right now I'll try to be more thorough when I get home, but an important thing to note about the Kurdish people is they have never been a unified people under one state. They've faced oppression by basically every empire that has existed in the area, so they're division today can be seen as a modern continuation of that. A common Kurdish saying "No friends but the mountains" reflects this, as well as the mountainous region that comprises a majority of 'Kurdistan', i.e. not much oil. But if I do remember correctly Iraqi Kurdistan produces a relatively large portion of Iraqi oil. Unfortunately a lot of Kurdish groups resort to violence, and are grouped with terrorist groups in the region, inhibiting the compliance of governments, especially Turkey. Your last question is more policy all science though. Check this quality post
They never were together. Kurdistan was never a country or unified autonomous region.