When I cough, I can't seem to breathe properly for couple of minutes. Is this normal?

Not usually. It can be a sign of asthma, which is characterized by bronchospasm, or the tightening of muscles around the airways. When a person coughs or laughs too much or exercises quickly without warming up, these muscles tighten, and can cause some temporary difficulty breathing.
No... Any cough associated with shortness of breath needs to be evaluated by a doctor asap so diagnosis can be made and treatment initiated. Testing, including pulmonary function testing, may be needed to diagnose these symptoms. If you smoke, you need to quit immediately and your doctor can help with this also. These are serious symptoms which require a doctor's evalution asap!