Is asthma transmissible?

Not really. Asthma is not an infectious disease and so is not transmissible as such. Viral infections or colds that worsen asthma are transmissible or infectious. Asthma often may have a genetic component and so does run in families.
No. Some infections can cause asthma, but asthma can not be contagious.
Not really... Asthma is a disease characterized by recurrent episodes of bronchospasm and airway inflammation and is not contagious. However, upper respiratory infections are a common trigger of asthma and these are contagious.
Sort of. Asthma is a polygenetic disease. So, you get it from your parents. 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.