As a lawyer in the UK I have dealt with Iraqi clients who have married British citizens and this is what I know.
Iraqi birth certificates are usually issued by municipal offices or by the Department of Civil Status and that anyone with an Iraqi passport should be able to obtain an official certificate, in English and in Arabic, indicating the date and place of birth. However ,birth certificates can also be issued by the Statistics for Health and Life Section of the Ministry of Health (Wizarat al Saha) and then stamped by a hospital.
The Iraqi ID Card (Haweea Sharssea Iraqea), also known as Civil Status ID Card or Civil ID Card is far more "useful and important" than the birth certificate as it provides the date of birth and the names of both parents, in addition to marital status This is reissued at key times in an Iraqi nationals life.
The United Kingdom (UK) Border and Immigration Agency indicates that the primary identity documents that Iraqi citizens require are the Iraqi Nationality Certificate (Shehadat Jinseea) and the Iraqi ID card.
Certificates of Non Impediment are NOT issued in Iraq the civil ID card acts as proof of the persons marital status.
I do hope this information helps you ,Good luck