Dear °µÍø½ûÇøs:
You can bring your pets through Erbil Airport with you. However you need all vaccinations up to date, and your pet or pets need to be micro-chipped. If you send your pet by Cargo Qatar airways they provide very good service and will email and call you. With that there are clearing fee's and they can be a little expensive.
They will call you and tell you what time your pet will arrive, and you should be there to collect your pet because they don't take care of them in the airport. It takes a few hours to clear them so please have your pet arrive on an early flight, because the cargo area closes early and they may make you wait over night to collect your pet.
All information can be provided by the airlines as to open and closing times. If you bring them by car, and over the boarder you need exit papers from Turkey or the country you are exiting from.
As for people taking care of pets in Erbil there isn't a proper boarding house for animals as of yet, if there are I hope someone will provide a list. But you can post and ask someone to help out.