From what main artery does the coronary artery branch off?

The main artery would be the aorta.

In response to your second question, the coronary arteries take the oxygen-rich blood from the heart to send to different parts of the body. As a result, if these arteries are blocked, then your body would be lacking oxygen-rich blood, leading to muscle and brain death and then your eventual deah.

The aorta.

The coronary arteries are the first to branch off ----- they innervate both sides of the heart. They are important because your heart is made of cardiac muscle (all muscle requires blood flow for contraction), and without the coronary arteries supplying your heart with blood for its own muscle, then the heart would never contract; if blood flow is stopped, many things may happen including hypoxia (Maybe heart attack), and eventually death.

The coronary arteries do not extend from the pulmonary circuit (your pulmonary arteries bring used blood to the lungs, and back to the heart through a pulmonary vein), that is a closed loop. Once newly replenished blood is in the left atrium, it gets pumped back out through the aorta. This is where the blood destined for your coronary arteries branches off to the muscle lining your heart.

the ascending aorta

The aorta.