Civil war between Sunnis on one side versus Shi'ites & Kurds on the other
is probably inevitable, and may already be underway. Since the Sunnis have failed to join up with the new government, following traditional Muslim policy, they must be exterminated. Those are the rules over there.
If the Iraqi government wants the continued support of the USA, including both money and firepower on demand, it must compromise. And it needs that support.
The oil was nationalized decades ago. There are no private oil wells in Iraq. The entire oil industry belongs to the nation and has for a very long time.