The Standard Models (SM) of particle physics and cosmology have been great successes so far, but various observational and theoretical hints point towards new physics beyond them. In this thesis, we first briefly discuss these shortcomings, including puzzles for the initial state of the early universe and how they can be solved via Cosmic Inflation. We then focus on constructing microscopic models for inflation which are theoretically natural, Effective Field Theory (EFT) controlled, and observationally consistent, while also looking for possible new signals. We develop a supersymmetric (SUSY) bi-axion model of high-scale inflation, in which the axionic structure originates from gauge symmetry in an extra dimension. While local SUSY is necessarily Higgsed at high scales during inflation we show that it can naturally survive down to the ~TeV scale in the current era in order to resolve the electroweak hierarchy problem. In the face of improving constraints on the tensor-to-scalar ratio, we also investigate inflation at lower energy scales via the very well-motivated mechanism of Hybrid Inflation. We construct a technically natural and EFT-controlled model for this, “Twinflation”, incorporating a discrete “twin” symmetry.

If a SUSY extension of the SM does survive down to ~TeV scales, although not yet observed at the collider searches so far, it may have structures giving rise to novel Long-Lived Particle (LLP) signatures. LLPs also feature in a variety of other new physics scenarios. We show that future electron-proton colliders, forming an interesting hybrid of leptonic and hadronic colliders, can probe LLPs with soft decay products and very short lifetimes, thus offering a complimentary reach into the new physics parameter space.