Of the most important open questions in cosmology and perhaps all of physics is the observed late time cosmic expansion. The accelerating expansion rate of the universe, driven by vacuum energy, presents a profound challenge. Theoretical predictions for the value of this vacuum energy are discrepant by an astonishing factor of $10^{120}$ relative to the observed value necessary to drive the current levels of expansion, highlighting a significant gap in our understanding of fundamental physics. This thesis aims to explore and constrain alternative theories of gravity that extend beyond General Relativity (GR), which have since arisen as a potential solution to this cosmological conundrum. This thesis presents a detailed examination of the impact of modified gravity on the emptiest structures in the universe, cosmic voids. It evaluates the robustness and usefulness of void statistics in different gravity models, and addresses the implications of statistical noise on simulation derived Fisher forecasts for large-scale structure surveys. First, we investigate the effects of cosmic scale modifications to general relativity on the dynamics of halos within voids by comparing N-body simulations incorporating Hu-Sawicki $f(R)$ gravity with $\Lambda$CDM models. By analyzing the radial velocity statistics within voids classified by size and density-profile, we uncover significant differences in halo motions within small "$R$-type" voids between $f(R)$ and $\Lambda$CDM cosmologies. We develop an iterative algorithm to solve the nonlinear fifth force equation in $f(R)$ gravity which allows for a sensitive characterization of the Chameleon screening mechanism in voids and highlights the distinct behavior of the fifth force in these environments. Next, we compare void size and clustering statistics across $f(R)$, $nDGP$, and general relativity using N-body simulations. Our study emphasizes the importance of using mock galaxy catalogs rather than dark matter halos alone for more realistic void identification. Despite enhanced void radial velocities and velocity dispersions in modified gravity models, the redshift space void quadruple moments derived from mock galaxy tracers remain remarkably similar across different gravity models. We employ the Gaussian Streaming Model to accurately reconstruct $\xi_{2}$ and demonstrate the void quadruple as an unbiased estimator of the redshift space growth rate parameter $\beta=f/b$ in modified gravity theories. Expanding our scope from cosmic voids, we turn to another crucial aspect of cosmology and particle physics: the nature of massive neutrinos. A number of current and upcoming cosmological surveys aim to tightly constrain the neutrino mass sum which in combination with ground based particle experiment observations could decipher the neutrino mass hierarchy. In the final chapter of this thesis, we address the challenges posed by statistical noise in simulation-based Fisher forecasts for large-scale structure surveys. We reveal how noisy numerical derivatives, due to a finite number of simulations, bias the Fisher forecast, resulting in overly tight marginalized constraints. This effect is particularly pronounced for parameters like neutrino mass, where higher-order differentiation schemes are commonly used. We provide a statistical analysis to characterize these biases and propose methods to recover noise-free Fisher constraints, ensuring accurate assessments of upcoming survey capabilities. In conclusion, this thesis advances our understanding of the effects of different theories of gravity on cosmic structures, refines void statistics methodologies, and offers solutions to mitigate the impact of statistical noise on cosmological forecasts based within simulations, thereby enhancing the precision and reliability of future large-scale structure surveys.