As I understand it, Iraqi Shiites already dominate the current government and have ethnically cleansed the Sunnis from Bagdad. All Iraqi Shiites are pro-Iranian. Is this not true?
Not at all. If anything, there are more Sunnis in Baghdad than there are Shiites. The Shiites dominate the southern cities such as Basra, Najaf and Nasariyah. Baghdad is a mix of everyone, including Christians.
And NO, not all Iraqi Shiites are pro-Iranian. All pro-Iranian types here are Shia, but I personally know many Shiites who don't want any government here that resembles that of Iran's. These people all lived in a secular society for decades and they still lean very much that way. That's why Allawi got the most votes in the March elections - he campaigned on secular government principles.
There are a lot of the pro-Iranian scum here and most of them follow that vermin Muqtada al Sadr. But the majority of Iraqis despise his militant tactics and just want peace.