Interphase nuclear rupture is a known driver of DNA damage and may ultimately lead to genetic instability promoting cell dysfunction and cancer progression. As the structural support of the nucleus, the nuclear lamina minimizes rupture by protecting against mechanical deformation. Nonetheless, nuclear envelope breaks become more frequent when a nucleus undergoes severe deformation in response to environmental constriction or applied pressure. Curvature in particular has been identified as a major determinant of such breaks, with sharp probes and constricting pores perturbing the integrity of lamin filaments and generating more nuclear rupture. Despite this known association, characterization of nuclear lamina dynamics while subjected to Gaussian curvature and investigation into the potential consequences of intracellular organelles imposing such curvature onto nuclear and cytoskeletal filaments are lacking. Here, live imaging and nuclear rupture quantification across a range of physiologically induced curvatures suggest that Lamin-B depletes immediately upon adoption of high curvature, which contributes to enhanced rupture at these locations, whereas Lamin-A requires a critical strain rate to dilute. Consistent with these findings, the imposition of high Gaussian curvature onto the nucleus by small, stiff lipid droplets is sufficient to dilute Lamin-B filaments and trigger nuclear rupture in multiple cell types. Frequent rupture caused by droplets contributed to downstream dysfunction by initializing mislocalization of DNA repair factors and elevated DNA damage. Beyond their interactions with nuclear lamins, small lipid droplets also displace actin filaments and can impair cytoskeletal processes in macrophages, particularly phagocytosis. Notably, despite overall impairment in uptake, more severe disruption to the cytoskeleton resulting from higher lipid droplet accumulation leads to increased myosin activation and more phagocytosis. Finally, a live-cell ChReporter identified ongoing genetic instability within a cell and underscored how genotypic alterations result in tangible differences in phenotypic characteristics such as morphology, motility, and proliferation. Altogether, this thesis utilizes biophysical principles to elucidate cell behaviors and consequences associated with physiologically induced nuclear rupture and disruption of the cell cytoskeleton.