Is asthma communicable?

No. It is not; asthma is the body's response to allergens or irritants in the air. Individual's responses to these stimuli vary widely, but they are not shared.
Sort of. Asthma is a polygenetic disease. So, you get it from your parents. Also, triggers for people who have "underlying" asthma certainly include transmissible disease such as viruses that cause URI. Asthma involves inflammation of the lungs, which is constantly being suppressed during times of good control. With an infection, the suppression mechanism is released and "asthmatic inflammation" follows.