What is the source of energy responsible for Earth movements (rotation, translation, etc.)?
Energy is not required for movement. You only need energy for changes in movement, such as some accelerations.
For the way we usually think about the Earth's movements, there is no energy source because it's already in motion. Orbital motion, in particular, is a kind which requires no additional energy... kind of like how it takes no additional energy to keep moving when you're slipping on ice.
If you want to get deeper than that, the energy which got the Earth moving in its current way did indeed come from somewhere. It came from the gravitational potential energy of dust as that dust coalesced into clumps, and later into planets. But we rarely choose to think that far in the past. Usually we say that the Earth has its kinetic energy, and the path of its orbit does not change that energy.