Is it possible to lose 20 pounds and still maintain a healthy lifestyle?

Yes. If you are 20 lbs overweight, then losing it (in a reasonable way) will definitely improve your health in many ways including (but not limited to): risk of diabetes, high blood pressure, sleep apnea, heart attack, stroke. But, check with your doctor first to make sure you are otherwise fit to start a diet/exercise program and to monitor any chronic disease you have, if any.
Yes. Eat healthfully. Try to eat at least 5 servings of vegetables & 4 of fruit per day. Increase nutritional fiber. Avoid refined sugar & junk carbohydrates. Avoid alcohol, sweetened juices, soda & caffeinated beverages. Aim for 7.5 to 8 hrs of sleep / night. Hydrate w at least 64 oz of water / day. Calories burned must > calories consumed. Do both cardio & weight training.