We study the stellar discs and spheroids in eight simulations of galaxy formation within Milky Way-mass haloes in a Lambda Cold Dark Matter cosmology. A first paper in this series concentrated on disc properties. Here, we extend this analysis to study how the formation history, structure and dynamics of discs and spheroids relate to the assembly history and structure of their haloes. We find that discs are generally young, with stars spanning a wide range in stellar age: the youngest stars define thin discs and have near-circular orbits, while the oldest stars form thicker discs which rotate ~2 times slower than the thin components, and have 2-3 times larger velocity dispersions. Unlike the discs, spheroids form early and on short time-scales, and are dominated by velocity dispersion. We find great variety in their structure. The inner regions are bar- or bulge-like, while the extended outer haloes are rich in complex non-equilibrium structures such as stellar streams, shells and clumps. Our discs have very high in-situ fractions, i.e. most of their stars formed in the disc itself. Nevertheless, there is a non-negligible contribution (~15 percent) from satellites that are accreted on nearly coplanar orbits. The inner regions of spheroids also have relatively high in-situ fractions, but 65-85 percent of their outer stellar population is accreted. We analyse the circular velocities, rotation velocities and velocity dispersions of our discs and spheroids, both for gas and stars, showing that the dynamical structure is complex as a result of the non-trivial interplay between cooling and SN heating.
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