The majority of the mass in the Milky Way (MW) is dark matter (DM). Understanding its distribution is essential to uncover its nature and role in galaxy formation and evolution. Traditional methods use measurements of stellar motions and integrate their orbits assuming a time-static and symmetric gravitational potential for the MW to infer its DM distribution. However, in the Gaia era, it is now well established that the MW is dynamically evolving and asymmetric. Symmetric-static models of the MW fail to accurately reproduce orbits, deviating from reality in $\leq1$ Gyr. We developed and validated sophisticated potential models based on basis function expansions that capture both the global evolution and distortions of the MW halo. We tested these models against state-of-the-art cosmological-baryonic simulations, and found that they preserve stellar orbits with position errors within 10% and properties such as energy and angular momentum with 2% errors over multiple orbital periods. We applied these models to test for the disequilibrium caused by the LMC in the MW. Our results show that DM subhalo interactions with stellar streams can be boosted by up to 40% near the LMC and 70% in diametrically opposite regions due to subhalos brought in by the LMC and the MW's response to the LMC. These predictions can help identify signatures of subhalo-stream interactions, providing constraints on the lumpiness of DM in the MW. Additionally, we investigated the effects of DM models on the matter distribution in the Solar Neighborhood. Self-interacting DM (SIDM) results in locally denser and more oblate halos compared to cold DM. This signature is imprinted on the acceleration field, with SIDM producing 10-30% steeper acceleration gradients than the cold DM model. However, satellite mergers can disrupt these gradients, leading to asymmetries in acceleration fields that persist over long timescales (3--4 Gyr). Our work shows how future measurements of the MW’s acceleration field will provide insight into its merger and formation history. Moreover, measurements of the acceleration fields from multiple galaxies will enable statistical analyses that can help constrain the nature of DM.