Yes, it does. But all exercises are not equal in this regard. Those using as many muscles as possible and allow you to handle the most weight are most beneficial.
Yes. Any amount of exercise will increase it some. The largest increase occurs if you engage in very high intensity exercise for at least 10 minutes during regular training.
No, the human growth hormone is naturally released into your body, the only way to stimulate this is with prescription medication. I suggest you talk to your doctor about your concerns.