Carrier Unlock Denied

If your cousin the US was the one who had the account and paid the phone off, then have them call AT&T and get it straightened out and have them get the unlock.  They would have a much better chance of success than a third party outside of the US.

Did you answer the question wrong by filling in the "yes" bubble? 

Your denial is based on you not having an AT&T phone number. Try resubmitting, and this time fill out the form correctly. 

The phone number on the sim is not accepted anymore so everytime I requested for an unlock I always used the IMEI of the phone. I was thinking, maybe, it had something to do with the length of the inactivity of the sim?

It has nothing to do with inactivity of the SIM.  Assuming you answered No to the question, and only entered the IMEI and still got this message, then the problem is with the unlocking site not working correctly.  A number of people have reported recently that they have gotten that message, even though they answered No to having an AT&T phone number.  Unless AT&T magically fixes the unlocking site, your main option is to call AT&T directly and talk to them.

Again, have you asked your cousin in the US who sent you the phone to get the phone unlocked?

@chLLaX

Please look at the picture I posted.

Did you fill in the bubble that says "no"? Or leave it at the default "yes"? 

"Yes" is indicating an active AT&T phone number which neither you nor anyone else has anymore. 

If you did not change the bubble to "no", that would explain the message you got and the denial would be correct.

If you filled in "No" and no phone number, the message is faulty.   Since you are out of the country you cannot deal with AT&T and the person who sent you the phone needs to do so for you.   They should have unlocked the phone for you before they sent it.

I'm actively having the same issue, however I and my family member do reside in the United States.  The answer my family member got from the AT&T representative is that since the phone has not been in service with AT&T for the past 6 months they would not be able to unlock the phone.  The phone was gifted to me. I had another carrier for years until, this year I opted to save money by bundling up some services with another carrier. When I attempted to switch the sims card, that is when I was notified the phone was locked.

I don't understand how a phone which legally belongs to me is now useless because AT&T wants me to get it's mobile service for 6 months before I can unlock it.  At this point I am going to dial up my attorney and determine which route we would like to take legally.  It's been a week with a phone that has been paid off years ago and no service.   I'm at my wits end.

@yoiseguia 

AT&T representative is that since the phone has not been in service with AT&T for the past 6 months they would not be able to unlock the phone.  The phone was gifted to me.

Since the phone was a gift, that means someone else purchased it, as far as anyone is concerned you are not the legal owner The person who bought it is.  

AT&T maybe willing to help the original purchaser unlock the phone, but they won't even talk to you, you're not their customer.

Since you are being told the phone has to be used on AT&T for 6 months, I suspect the phone purchased is an AT&T prepaid phone that has not been activated or used on AT&T.  Prepaid phones are sold at a discounted price and therefore the FCC allows AT&T to lock those phones for up to 12 months.  ATT only requires 6 months.