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The ball hits the ground where some energy is dissipated to heat and friction. While traveling back up, the ball's energy is losing kinetic energy while gaining potential energy (conserved energy). With each successive bounce, it dissipates more energy to friction, air resistance and heat.
Beside this, what happens to the energy of the ball when the ball stops bouncing?
The short answer is: A ball stops bouncing because it has lost all its energy. Upon hitting the ground, some kinetic energy is lost, because it is converted to other forms of energy, mostly in the form of thermal energy (from friction and heat released) and even possibly sound energy.
Beside above, how long does it take for a ball to stop bouncing?
The answer, as far as I can tell, is yes (well, maybe). Since each bounce has half of the initial velocity of the previous bounce, the time between each bounce also halves. The first bounce might take 1 second, the second takes 0.5 seconds, the third 0.25 seconds, etc.
The law of conservation of energy implies that a bouncing ball will bounce forever. When you drop it on the floor, it changes some of its energy into other forms, such as heat, each time it hits the floor.