treadmill

A treadmill will burn calories and improve your cardio/vascular condition. It's good for you. Can you lose weight on a treadmill. Yes, of course you can. As to your third point the calorie intake is more important than anything. On average you'll burn 6-10 calories a minute doing something like jogging on a treadmill. So if you jog 30 minutes in the really sweating range you'll go through around 300 calories. You can erase all that with a simple big spoonful of peanut butter. You can always, always, always out eat your exercise. That's why having a calorie diary is important.

Exercise for your health and monitor your calorie intake for weight control. Both are important but I would say that you calorie intake is 75% of the deal.

EDIT: Your body burns through calories on it's own to stay alive. Depending on your height/weight and activity level I'd guess you're probably burning through 1500-2000 calories without ever visiting a gym. You don't have to do hours of cardio each day to lose. Personally I wouldn't do more than 30 minutes at a time unless your training for something like 5k or marathon.

Second EDIT: You can't tell your body where to drop fat from. Consider all fat (bell, thigh, back, arms) just one big bucket of fat. You have to empty the bucket as a whole to see specific areas go down in size. There is no such thing as spot reducing fat. Your body holds on to it according to it's own plan. That can not be changed. You can only raise or lower the total level.