Can I still lead a normal life if I have asthma?

Yes... There are effective asthma treatments and most patients are symptom-free between exacerbations and there are often long periods of time between exacerbations. There are many athletes, even professional and olympic-level, who have asthma. So you can live a normal life with asthma!
Yes. You may have to make some concessions to asthma like not smoking, taking medications to control asthma, taking medication before exercise to prevent bronchospasm, and even finding a new home for your cat. That still fall into my definition of "normal life".