The large datasets of precise cosmological observations set to arrive in the next decades present challenges in data science, machine learning, and model building aspects of cosmology. Following a primer on cosmic inflation, accelerated expansion, structure growth, random fields, and cosmological parameter inference, this dissertation is a collection of probabilistic and statistical arguments intended to answer the following questions. First, does today's available data have enough constraining power to non-parametrically answer fundamental questions about a dynamic dark energy? Using Gaussian processes to model the dynamics, we conclude that while there is enough power for non-parametric modeling of dark energy and its phase space, the constraints are not strong enough to answer questions posed by the Swampland Conjectures in quantum gravity. Secondly, is inflationary cosmology limited or affected by the Refined Swampland Distance Conjecture? Using the statistical properties of Gaussian Random Fields and inspired by heuristics from supergravity, we argue analytically and numerically that the stochastic nature of inflation would most likely place today's Universe where the EFT describing inflation breaks down. This means we should not expect to describe the inflationary far past and the present with the same EFT. Third, turning to cosmological parameter estimation, what does the cosmological data we have really measure? Using information geometry, we demonstrate robust linear and non-linear techniques for finding the directions in parameter space that the dataset independently constrains with no degeneracy or correlation. This will be key in breaking degeneracies and maximizing constraining power. Lastly, as data vectors of summary statistics in cosmology have grown exceedingly long, are we making the most out of the data we have? Based on the simulation based inference framework, we use information theory and real examples to identify the optimal dimensionality reduction techniques that boost constraining power by as much as 70%. In the era of precision cosmology this will be key in maximizing the constraining power. The results in this dissertation are a demonstration of the usefulness of good probabilistic and statistical arguments in solving the theoretical and data science challenges of cosmology.