Is the spin and charge of an atom a quantum or classical concept? I have no idea whether these properties of an atom fall under quantum or classical physics, or perhaps both. Some clarification would be helpful.
 A: The charge of an atom is defined by its constituent number of protons/electrons and local fluctuations in their density distribution which cause instantaneous dipoles, unless we are talking about ions which have a permanent charge. Charge is a classical concept that has real meaning in classical physics and can be described in various fields (magnetic/electric) using electrodynamics and Maxwells laws. Even simple dipole-dipole interactions between non-ideal gases are accounted for albeit in a limited way) by classical Van der Waals theory. At a very simple level: you do not require a quantum description to account for the charge of an atom. Of course in order to obtain any real data one would need to use ab-initio quantum calculations such as Hatree-Fock or various Variational methods and other assorted computational methods such as Density functional theory but this focuses on the density operator while the former uses the wavefunction. The concept of charge is as a parameter which can be fine tuned to obtain the useful detailing of the system under investigation i.e coupling interactions/repulsion terms/correlations... etc....
Spin on the other hand is a little different. There are two types of spin:
-the intrinsic spin of a particle (is defined by what type of particle we have) 
and, 
-the orbital spin. 
These two types of spin are crudely described by classical celestial physics and are likened to planetary motion, the earth orbiting the sun yet rotating about its own axis. In so much as an electron can be modelled by a planet (Bohr theory) then yes spin does have a classical interpretation. 
In order to properly describe the spin-orbit interaction you will need relativistic QM as Schrodinger wave mechanics doesn't account for the intrinsic coupling. This is evident in the singlet/triplet states of Helium for instance. The Dirac equation is the equation of choice in canonical quantum mechanics, while there are other quantum field theory approaches you could take to use a Lagrange approach which I leave to someone more adept than I (at three O'clock in the morning at least ! :) 
In summary: Charge is described by both, but the "true" picture short of a hard shell particle is left to QM. While spin has only tentative parallels with classical mechanics and in general cannot (as far as I am aware) be described to any useful detail by classical mechanics or introductory QM, instead requiring a second quantisation or relativity in some form. 
Cheers. 
