Supernova remnants (SNRs) offer the means to study supernovae (SNe) -- the explosions of stars at the end of their lifetimes -- long after the original explosion and provide a unique insight into the mechanisms that govern these energetic events. In particular, the deviations from spherical symmetry observed in many SNRs can be compared to predictions from recent 3D simulations to further our understanding of SN physics. In this dissertation, I use archival data from the Chandra, XMM-Newton, and ROSAT telescopes to study the X-ray emitting material within supernova remnants to help constrain the physics of SN explosions and SNR expansion. I begin by investigating the relationship between bulk ejecta asymmetries and neutron star (NS) kick velocities in a sample of 18 SNRs. NSs -- the compact objects leftover from certain types of SNe -- have been observed to be kicked to hundreds of km/s velocities, thought to result from a conservation of momentum-like process with either asymmetric ejecta or anisotropic neutrinos emission. I measure SNR asymmetries using the power-ratio method (PRM; a multipole expansion technique) and compare these asymmetries to the kick velocity of each remnant's neutron star. Although I find no correlation between the magnitude of power-ratios and NS kick velocities, I do find that NSs preferentially move opposite the bulk of ejecta in 5 out of the 6 SNRs with robust NS velocity measurements. This result is consistent with the theory that NS kicks are a consequence of ejecta asymmetries as opposed to anisotropic neutrino emission. I then examine the morphologies of individual elements in the youngest known core-collapse (CC) SNR in the Milky Way, Cassiopeia~A. The same asymmetric explosion mechanisms that generate NS kicks should also affect the distributions of elements synthesized in explosion. I apply the PRM to maps of individual element emission and find that the NS in Cas~A is kicked in a direction opposite to the heaviest elements (e.g., Ar, Ca, Ti, and Fe) and these elements exhibit more asymmetric morphologies than lighter elements (e.g., O, Si, S). These results further support the theories that asymmetric explosion mechanisms are an important component in this SNR explosion, and that NS kicks arise from interactions with the ejecta. Subsequently, I turn to investigating the presence and location of recombining plasma in the SNR W49B. Recombining plasma is overionized -- the ions are in higher ionization states than the electron temperature would suggest -- and is thought to result from rapid cooling of electrons through either adiabatic expansion into a low-density medium or thermal conduction with dense gas. I perform a spatially-resolved spectroscopic study of {\it XMM-Newton} data across 46 regions and find that recombining plasma is present throughout the entire SNR, with increasing overionization from east to west. Notably, in contrast with past works, I do find evidence of overionized plasma in the east where the SNR is interacting with dense molecular cloud material, suggesting a thermal conduction origin. I determine that small-scale thermal conduction with small ($\lesssim$1~pc) embedded, dense clouds is a plausible origin for rapid cooling in these regions. In the future, I will continue to study X-ray ejecta emission in SNRs in order to constrain SNe explosion mechanisms and progenitor scenarios. I will expand my thesis work, using newer X-ray observations to obtain a larger sample of NS kick velocities and SNRs. In addition, I plan to systematically measure the abundances of elements in a large sample of SNRs, comparing these results to the predictions of 3D SNe models. In particular, I will make use of upcoming X-ray telescopes that will have much improved spectral resolution, using these observations to generate precise spectral fits to more accurately measure precise plasma properties in X-ray SNR spectra.