Do I have nerve damage 1 week after FUE?

If you are having issues after surgery, especially with "pain" you should contact your doctor.  Some nerves are always affected when you are having surgery. 

İt is a mild , transient ,dermal plexus nerve damage. We can observe that much pain on donor site %15-20 of my patients after 7-10 days after surgery . That pain is because of skin dermal nerve plexus injury and temporary , mostly disappear in 1 week . 

Yes, what you are describing is a nerve injury. The FUE transects many small nerves but hopefully not the deep nerves which would be a real problem. Wait it out 2-3 months and hopefully if it is a small nerve, it will get better