In our atomic physics class, we saw that the spin-orbit coupling term arises from the scalar product of the magnetic moment of the electron (proportional to its spin), and the magnetic field created by the nucleus rotating around the electron. But this implies that in the electron's reference frame, it still sees it's own spin? How is this possible?
Spin is an intrinsic property of quantum objects that, unlike a particle's orbital momentum, does not depend on the frame of reference you are considering. Another intrinsic quantity of that kind would be charge, which is also just a fundamental number you assign to a particle, no matter its state of motion.
One possible source of confusion when talking about spin is the fact that it is similar to classical or orbital angular momentum in many ways, e.g. it follows the same mathematical structures (for example addition of angular momenta). It is, however, still different. Electrons, and elementary particles in general, are, to the best of our knowledge point particles. Our classical intuition of something "spinning around its own axis" requires extended objects, a notion which definitely breaks down at the fundamental level of quantum mechanics.