I've been flying in and out of Iraq for the last 5 months, into Baghdad, Balad and Mosul. It's risky flying in there, however you are well safe above 18000'. We needed an additional insurance for operations in and over Iraq, which is valid for day operations only, so I don't think airlines are overflying Iraq at night.
There's no overflying over Baghdad airport/city for airlines which are just over flying Iraq, the airway takes you well away from Baghdad. So even if an airline would loose the cabin and descended to 14000', you're ok, because you are away from the hotspot. You're just flying over farm fields, and no cities. Closest you would be to Baghad is about 25Nm's.
Remember the airbus of DHL got shot at around 8000' when departing Baghdad. These guys were getting used to flying in there, and decided to fly straight in and depart straight out, day after day. This is a no no, as you spiral in and out of there, leaving the spiral at FL150 to proceed enroute. This procedure goes for all airfields in Iraq. Safe margin of 5Nm's of the aiport in most airfields, Baghdad requires spiral in and out over the airfield perimeter, which requires steep turns
If one would decide to divert (going southbound), I would recommend to divert to Kuwait. Exit Iraq via TAMIM then proceed FALKA to KUA.
Hope this helps....