Can I still become a firefighter if I have asthma?

Bad idea. While it is a noble calling, you will be exposed to dense smoke, intensely hot dry air, dust, and ashes all of which can trigger your asthma and may lead to a life threatening attack. Since others are relying on you to be both a rescuer and a part of the team, getting taken out in the middle of action might make you a liability. Police work, medicine, etc are all still available...
Absolutely. Just like asthmatics can be olympic sprinters or swimmers, or anything else they want to be. The key is asthma control, not whether you have asthma or not. If a person's asthma is under excellent control, under the supervision of an allergist, then there should be nothing you can't do!