Is it possible to have asthma and not ever have asthma attacks?

Yes. If you control your disease adequately you may have very minimal symptoms and live a normal life. It is not always possible to do so (i don't want to shame anyone with difficult to control asthma who is adherent to their regimen), but the vast majority of asthma sufferers never need to be hospitalized, and many can be symptom free.
Yes. It is possible to wheeze and have bronchial obstruction without feeling out of breath; in fact, this is common in chronic, long-standing asthma.