If there is a rotating magnet in an empty space and there is no outer field acting on it. Rotating in such a way that after half rotation magnet's N pole will be in the place where magnet's S pole was. I imagine that spin of the electrons inside magnet are following this rotation. Axis of rotation of electrons is perpendicular to the line connecting S and N pole of the electron. Do electrons inside the matter change their spin in "steps" or is it continuous movement? Are electrons trying to counteract this movement (like when you are trying to change rotation axis of macroscopic object)?
A rotating magnet can be handled as an "orbital" angular momentum problem, no spins needed, but nothing stops you from using the usual spin-orbital coupling to calculate the total magnetic moment of a realistic solid state system. What matters for the interaction of the magnet is its total magnetic moment. Since angular momentum is quantized, the magnet can only change its angular momentum under the influence of an external field in a quantized way. There is no heating involved when such changes occur, the magnet will simply rotate faster or slower. Yes, the Einstein-de Haas effect is quantized.
Now, if you are asking about spins in a magnetic field (whether it is generated by the spins themselves of external), a single spin can only be up or down. Since the energy difference between a spin in up vs. down state is small compared to $kT$ in magnets at room temperature, magnetization has to be treated as a thermodynamic problem, in which the average magnetization is a thermodynamic average over a macroscopic number of spin states. Unless we go to really small systems with only a handful of atoms at low temperature or we do magnetic field resonance with many spins in a correlated state, the quantum mechanical properties of a magnetic system won't show up as an obvious effect.