I'm 25, can I already be getting COPD or emphysema from smoking?

Maybe. If you are a heavy smoker emphysema can occur at an early age (less than 35). There is also a genetic disease called Alpha one antitrypsin deficiency that can cause emphysema to develop at any early age. If concerned you should seek medical evaluation.
Possibly. There is a genetic disorder known as Alpha 1 antitrypsin deficiency that could result in emphysema at a very early age. If this is not present, it would be very uncommon to have emphysema in your 20s.