The discovery of the Cosmic Microwave Background marked a pivotal moment in Physical Cosmology, propelling the field into an era of unprecedented precision in describing the Universe. Over the past two decades, rapid advancements in spectroscopic surveys have enabled the probing of matter distribution on cosmic scales with unparalleled accuracy and temporal resolution. A detailed understanding of this distribution is crucial for exploring the Universe's expansion history, estimating the energy content in baryonic and dark matter, and revealing the nature of the elusive dark energy, which is thought to drive the Universe's accelerated expansion. Studies of the large-scale structure of the Universe, when combined with independent cosmological probes, allow us to test leading models of cosmic history and the nature of gravity on cosmological scales.

However, the rapid progress in cosmological surveys has introduced new challenges. The increasing volume and statistical complexity of modern galaxy surveys have pushed the limits of our ability to simulate the intricate nonlinear gravitational evolution necessary for testing cosmological theories. Furthermore, the development of novel observables has become essential, expanding our ability to extract valuable information from galaxy distributions while testing the limits of advanced modelling techniques.

This work addresses these challenges by presenting advancements in accelerating full $N$-body simulations, achieved through improvements in matching low-resolution simulations to higher-resolution ones. Moreover, it enhances the approximate mock pipeline by developing models for the sub-resolution scales of these simulations, thereby relaxing the resolution requirements of large approximate simulation boxes. Furthermore, this thesis explores the use of Cosmic Voids as probes for Baryon Acoustic Oscillations analysis, offering insights into the large-scale structure of the Universe. Additionally, novel simulation-based inference techniques are explored, providing a pathway to circumvent the hurdles associated with theoretically modelling alternative observables in the cosmic web.

Through these contributions, this study enhances our understanding of the Universe's large-scale structure and strengthens the foundations of Physical Cosmology by bridging the gap between observational data and theoretical models.