Absolute zero for atoms is the lowest possible energy state. The electrons are "moving", but they can't lose any more energy.
Absolute zero is a temperature. A single atom doesn't really have a temperature. Temperature is a property of a collection of particles, atoms or whatever. So consider an ideal gas of many single atoms. The temperature of that gas is a measure of the average kinetic energy of the atoms. At absolute zero, that average kinetic energy would be zero, which would imply that all of the atoms are no longer moving.
Yes. As has been pointed out, temperature is an averaging of energies of moving particles (of matter). Temperature is another physics concept that will be needing revision, and the list is getting longer fast. Can thermodynamics be far behind? Matter at the ground state, perfectly insulated and neither absorbing nor emitting photons of any energy, is nevertheless interacting with the Higgs field, which is, by the way, a much more energetic interaction than whatever happens to the same atoms with massless photons. Good one or even two or three. The range of possible temperatures (through infinity, negative infinity) is old hat. They even taught that to me in the 1970s.
Just to deal with the electron part of your question, electrons definitely do not stop moving in an atom at absolute zero. The atom will be in its Ground State, which means the electrons will be in the bottom energy level permitted to them. But in this bottom level they are still in motion - or at least the particle analogue of these wave-particle entities is effectively in motion. This is one of many examples of so-called zero point energy, i.e. residual energy that cannot be extracted.
While absolute zero is not attainable, a group of atoms/molecules can have a temperature very close to it. Google BEC which stands for Bose-Einstein Condensate. It might be necessary to google the full name rather than BEC. The properties of such a group of atoms/molecules is considered to be experimental confirmation of the Uncertainty Principle. The atoms/molecules in a BEC seem to lose their individual identities because the momentum of each is very close to zero (Id est: Very precisely known), making the locations less precise. Each atom/molecule seems to occupy a volume much larger than normal & the locations of each atom/molecule seem to overlap. Contrary to the belief of many people, the Uncertainty Principle is not a statement about some limitations on measurement technology. It is a claim that increased precision relating to location requires decreased precision relating to momentum.
