I didn't realize Chaldeans still existed. I remember reading about them in the Bible. I even found Basra (Iraq) in the Bible. It's all amazing to me.
The Iraqi Christians were well protected by Saddam. Now, under US occupation, and Sistani's Sharia laws their days are numbered. What a sham!
The religious situation is very complicated because of ancient divisions (5th-century A.D.) between Orthodox, Nestorians, and Monophysites, and later Catholic efforts to reunite Christians in the Middle East with the Catholic Church.
The Chaldean Christians are in union with Rome and for their liturgy use a form of the Syriac language (close to the Aramaic language used by Jesus). Before their union with Rome they were considered Nestorians.
The Assyrian Christians are not in union with Rome. Apparently they use the same liturgical language as the Chaldeans (it looks like the Chaldeans are basically a segment of that church which decided to reunite with Rome). They are Nestorians.
There is a Syrian Catholic Church with members in Iraq--a bishop of that church was recently kidnapped in Iraq but soon released. They apparently use a different form of the Syriac language in their liturgy. There's also a Syrian Orthodox Church (in union with the Greek Orthodox) but I don't know if any of them are now in Iraq.
The Mandeans are Gnostics (neither Christian nor Muslim). The Yazidis seem to be an offshoot of Islam. I think they have been called "devil worshippers" but that may be a misrepresentation of their actual beliefs.
I don't know anything about the Shabak.