Q: unlocking iphone 5s

Did you buy the iPhone 5s from Apple or T-Mobile? And if the phone is locked to the T-Mobile network you will have to contact them to unlock the device and ask what their unlocking policy is.

Which T-Mobile?

Well, Strange, Contact th carrier and try to figure out why your contract free device was locked to their service; they have the final say

An Apple Store can sell a locked iPhone as well as unlocked ones. The customer must specifically ask for an unlocked iPhone. To the best of my knowledge, only in the US are T-Mobile model iPhones unlocked.

Regards.

You will need to contact tmobile. They will unlock after 40 days on a contract.

Apple in the US now sells 2 unlocked phones: one with a T-Mobile SIM and one without a SIM ("SIM free").  They are both GSM models, so if you have GSM in Iraq, either should work. 

Even if you signed up for a prepaid plan with T-Mobile in the US, you should be able to remove the T-Mobile SIM, insert a local Iraqi SIM, and sign up with an Iraqi carrier.

This all assumes, of course, that you bought an unlocked phone from Apple.

They are wrong and giving you the runaround.Apple does NOT unlock phones.

Where did that quote (ORIGINAL CARRIER:) come from?  Was that on your iPhone?

You have wasted your money. Those IMEI sites are usually a scam.