Hopefully others will weigh in too. Vigorous walking and jogging use about the same amount of calories per mile, it is just a little faster w/jogging.
As far as lung capacity, training usually helps improve capacity, up to a point. Mostly training improves your muscle's ability to get and use oxygen. Since I don't know what the issue with your lungs is it would be hard for me say whether jogging will help you. You might pose that question to your Dr. or ask to see a pulmonary specialist. It is worth investigating.
Unless your Dr. says otherwise I suggest you keep up your run/walk program. That's how most of us runners started off. Just use your body as a guide and don't push it past the "just a little uncomfortable" stage.
Good luck, and keep us posted on your progress.