How can a resting object like earth have so much (infinite) energy to
attract objects?
Does this energy ever fades away, assuming no other external factors?
As @Alure explained to you, gravity is a force not energy. Energy can be stored in and retrieved from the gravitational field. When you raise an object the energy you expend is stored as gravitational potential energy. When the object drops gravity does work and the same energy is extracted from the gravitational field. For example, when your drone went from the ground to the height where it hovers, work was done to raise it and give it gravitational potential energy.
In a practical example, if I want to fly a drone, why does the drone
need to be constantly battling gravity? If my drone spends, let's say,
1000kJ to fly for 10 minutes, earth will spend the same amount of
energy to attract it, right?
The drone is constantly battling gravity in the sense that the force of gravity is trying to pull it down and the drone requires an equal upward force to keep it in the air. But the Earth expends no energy to attract it because it is doing no work (downward displacement of the drone) as long as the drone hovers. The Earth only does work if the drone falls in which case the work done by gravity will equal the loss in gravitational potential energy of the drone-Earth system.
So why does the drone need to continuously expend energy to keep it hovering?
Again as @Alure points out the drone does work (uses energy) to exert a force that propels air molecules downward (Work = Force x displacement of the air). According to Newton's third law the air molecules will then exert an equal and opposite (upward) force on the drone. If the upward force the air exerts on the drone equals the downward gravitational force, the drone hovers. If it exceeds it, it rises. If it is less, it falls.
Hope this helps.