Not usually. COPD is usually an acquired problem. However, if the causes are duplicated in the transplanted lungs (i.e. Heavy smoking) it could recur over years.
Hopefully not... COPD is usually caused by cigarette smoke so if you never smoke, and avoid second-hand smoke, you shouldn't develop COPD in the transplanted lungs. However, there are forms of COPD that are not caused by cigarette smoking and these could recur in transplanted lungs. Your doctor will be able to diagnose these other forms of copd.
No. COPD is not a disease that returns after transplant; some diseases like sarcoidosis can come back after a lung is transplanted and cause the new lungs to work poorly.