If we drop a bowling ball from one meter above the earth's surface, we are converting its gravitational potential energy to kinetic energy.
It is the ball and the Earth which have the gravitational potential energy not the ball alone.
When it hits the floor, it transfers its kinetic energy to the earth, so now the earth is moving directly downward at a larger (but imperceptible) speed.
Consider the ball and the Earth as an isolated system with both initially at rest and then let the ball fall down towards the Earth.
If there are no external forces acting on the system the total momentum of the ball and Earth system must be zero for all time.
As the ball moves towards the Earth it gains momentum but at the same time the Earth is moving towards the ball and gaining an equal amount of momentum but in the opposite direction.
Each also gain kinetic energy but the ball will gain much more kinetic energy than the Earth because the mass of the ball is so much smaller than the mass of the Earth.
If the ball and Earth stick together after they collide then the system has no kinetic energy.
The mechanical energy the system had becomes heat, sound and is used to permanent deform the ball and the Earth - it is an inelastic collision.
But let's say we dropped a hollow bouncy ball (eg a ping-pong ball) with the same mass. Now the ball can bounce back, elastically. Is this energy supposed to come from the compression and decompression of the inner gas molecules?
If the collision between the ball and the Earth is perfectly elastic then the mechanical energy of the system will be stored as elastic (spring) potential energy when the ball and the Earth deform during their collision.
Then that elastic potential energy will be converted to kinetic energy of the ball and Earth but now they are moving away from one another.
They would keep moving until their separation was the the same as their initial separation at which time they would be at rest relative to one another.
The sequence would then be repeated.