Does 62yo need physical for supervised walking, stretching+strengthening (bands/free weights+plate-loaded equip), swimming, water exercise/volleyball?

Yes. It is recommended for men over 45 and women over 55 to have a physical prior to engaging in an exercise program. The risk of heart disease becomes significant at those ages.
Not generally. If you have no current symptoms with exercise and have no known significant underlying disease you should be fine. If you develop symptoms see your doc.