Challenging $\Lambda$CDM: unraveling cosmic distances, dark sector phenomenology, and alternative primordial B-mode sources

The dominant Lambda Cold Dark Matter (LCDM) cosmological model, while remarkably successful, increasingly shows signs that it may not fully describe our Universe, as persistent tensions in expansion rates and structure formation remain unresolved. In this thesis, I challenge the LCDM paradigm using novel theoretical frameworks combined with rigorous numerical analyses. I demonstrate that the expansion-rate tension fundamentally reflects underlying distance disagreements, and that the Thomson scattering rate strongly restricts higher pre-recombination expansion rates without additional physics. Further, I present a novel cosmological model using a mirror dark sector and varying fundamental constants, revealing an observational degeneracy allowing significantly higher expansion rates while remaining consistent with data. Lastly, I show that detectable B-mode polarization signals can originate from phase-transition-induced gravitational waves, challenging the long-held view of primordial B-modes as definitive evidence for inflation. Collectively, this work advances our understanding of cosmic tensions and suggests clear avenues for future research.