Possible. Asthma can be triggered by environmental allergens such as pet dander and insects like roaches. Best to avoid known triggers. Triggers can be different in different people.
Absolutely. Cockroaches have been known to allergic triggers for decades. They are classically a problem in inner cities presumably because of the dense housing, potentially less than ideal cleanliness and the difficulty eradicating them. If you have asthma you should be seen by an allergists to identify your triggers and initiate appropriate environmental controls.