Is it possible for asthma to manifest only as a wheezing sound when taking deep breaths, without any coughing or difficulty breathing? would this be mild asthma?

Atelectasis is a term used to describe areas that are not fully expanded at rest. Taking a deep breath can open these & produce a wheezing sound. Asthma is a pattern illness where wheezing events are triggered by agents like infection/inhaled irritants/etc. You can have very mild asthma that never needs treatment, often called cough variant asthma. Your post is not specific to one label.