Why do electrons revolve around the nucleus? Why do electrons revolve around the atom's nucleus?
Where does it get the energy for the revolution?
Do the electrons stop revolving at absolute zero temperatures?
 A: Does it take energy to keep the Earth orbiting the Sun? It does not! It just takes gravitational force, which “costs nothing”.
Something similar applies to the obsolete Bohr model of the atom that you are thinking of. The electron has orbital energy, but it doesn’t constantly need more energy to keep going around. All it needs is the electrostatic attraction between the positively-charged protons in the atom’s nucleus and the negatively-charged electron.
In short, electrical attraction to the nucleus is what keeps the electron in orbit, and this attraction doesn’t cost energy.
You may find it odd that something can keep moving indefinitely without requiring a power source, but it is true. You have to add energy to the electron only if you want to change its orbit to be bigger, or rip it from the atom altogether.
The Bohr model ignores the inconvenient fact that an orbiting electron ought to radiate energy as electromagnetic waves. If this happened, the electron would quickly spiral into the nucleus unless you constantly gave it more energy to make up for what it was radiating away. It took quantum mechanics to explain why the electron doesn’t radiate: because it doesn’t actually orbit! Atoms are not actually like tiny electrical planetary systems.
Since electrons don’t require added energy to “keep going”, they don’t stop when the temperature falls, even to absolute zero.
A: 
Why do electrons revolve around the atom's nucleus?

The first approximate planetary model, Bohr model, is discussed in G.Smith's answer. There the electron is caught at the lowest energy bound state, given by the rules of the model.
The current model of the atom has no revolutions, because the term revolution is a classical mechanics term, and atoms belong to the quantum mechanical regime, where there are no orbits, but probable locations where the electron might be detected in an experiment, described by a probability curve. Here are the hydrogen orbitals:

These are calculated using the hydrogen wavefunctions, $Ψ$ which are solutions of the Schrodinger equation  and $Ψ^*Ψ$ gives the probability of the location of the electron around the proton at an (x,y,z,t), according to the Born rule

Where does it get the energy for the revolution?

In the Bohr model, the electron at a distance from the proton has a potential energy, part of it radiates away in photons, when caught at an energy level, and part of it  into the kinetic energy of the given energy level. 
This is true in the quantum mechanical model too.  It keeps to  this energy level unless it  interacts with some other particles ( interactions are another story) forever. That is why quantum mechanics is the theory of atoms, because it successfully models the observations at the small distances of atoms and molecules ( and more).

Do the electrons stop revolving at absolute zero temperatures?

Temperature is a classical variable used in thermodynamics, and thermodynamics emerges from the underlying many particle statistical model. A single particle has energy which contributes to the definition of temperature. Classically there could be a zero temperature in a material, this would not affect the hydrogen atom model of a proton with an electron in a -13.6eV orbital around it (electron needs this energy to be free of the proton). Here is a link to a discussion of absolute zero .
