The primary production of ATP comes from oxidative phosphorilation and the Citric acid cycle, but many think of glycolsis as an ATP factory producing a net of 2 ATP, although there is a net gain of about 36 ATP’s produced when oxygen is present (citric acid cycle). The primary molecules that produce ATP in glycolsis is the conversion of 1,3 Bisphosphoglycerate to 3, Phosphoglycerate yeilds an ATP and the conversion of Phosphoenolpyruvate to pyruvate yields an ATP these reaction are times by two but since there was 2 ATP invested to start the process you only yield a total of 2 ATP’s.
As for the energy stored in the ATP molecule: Phosphates "store" energy in molecules. Resonant structures of phosphates allow shifting of electrons and stabilization of adjacent phosphates which allow for high energy bonds. The energy is stored in the phosphate bonds.
Also the breakdown of ATP is not controlled by any one substance since the cell uses ATP as an energy currency any enzyme or molecule can use ATP and the breakdown is usually a hydrolysis reaction using water and yielding inorganic phosphates PPi or Pi. Through these hydrolysis reactions energy is given off and used by the cell to perform numerous reactions.