The standard model for particle physics has been extremely successful as a description of nature. Despite this success, there remain many unsolved puzzles both observationally and theoretically. In this thesis we explore a few ideas in search of beyond the standard model physics, especially we focus on the Higgs mass, magnetic monopole and vector dark matter.

In the first part of the thesis, we show that the Goldstone bosons of discrete symmetry can be parametrically lighter than otherwise expected. While non-linear realizations of continuous symmetries feature derivative interactions and have no potential, non-linear realizations of discrete symmetries feature non-derivative interactions and have a highly suppressed potential. These Goldstone bosons of discrete symmetries have a non-zero potential, but the potential generated from quantum corrections is inherently very highly suppressed. We explore various discrete symmetries and to what extent the potential is suppressed for each of them.

In the second part, we showed that in the early universe, evaporating black holes heat up the surrounding plasma and create a temperature profile around the black hole that can be more important than the black hole itself. As an example, we demonstrate how the hot plasma surrounding evaporating black holes can efficiently produce monopoles via the Kibble-Zurek mechanism. In the case where black holes reheat the universe, reheat temperatures above $\sim 500$ GeV can already lead to monopoles overclosing the universe.

In the last part of the thesis, we showed that vector Dark Matter (VDM) that couples to lepton flavor ($L_e$, $L_{\mu}$, $L_{\tau}$) acts similarly to a chemical potential for the neutrino flavor eigenstates and modifies neutrino oscillations. VDM imparts unique signatures such as time and directional dependence with longer baselines giving better sensitivity. We use the non-observation of such a signal at Super-Kamiokande to rule out the existence of VDM in a region of parameter space several orders of magnitude beyond other constraints and show the projected reach of future experiments such as DUNE.