The standard cosmological paradigm currently lacks a detailed account of physics in the dark sector, the dark matter and energy that dominate cosmic evolution. In this paper, we consider the distinguishing factors between three alternative models - warm dark matter, quintessence, and coupled dark matter-energy - and lambda cold dark matter (ΛCDM) through numerical simulations of cosmological structure formation. Key halo statistics - halo spin/velocity alignment between large-scale structure and neighbouring haloes, halo formation time, and migration - were compared across cosmologies within the redshift range 0 ≤ z ≤ 2.98. We found the alignment of halo motion and spin to large-scale structures and neighbouring haloes to be similar in all cosmologies for a range of redshifts. The search was extended to low-density regions, avoiding non-linear disturbances of halo spins, yet very similar alignment trends were found between cosmologies, which are difficult to characterize and use as a probe of cosmology. We found that haloes in quintessence cosmologies form earlier than their ΛCDM counterparts. Relating this to the fact that such haloes originate in high-density regions, such findings could hold clues to distinguishing factors for the quintessence cosmology from the standard model. However, in general, halo statistics are not an accurate probe of the dark sector physics.