Neutron stars in Low Mass X-ray Binaries (LMXBs) can accrete matter onto their surface from the companion star. Transiently accreting neutron stars go through alternating phases of active accretion outbursts and quiescence. X-ray observations during the quiescence phase show a drop in X-ray luminosity with the time in quiescence. This is also inferred as the drop in surface temperature or the cooling of accreting neutron stars in quiescence. Analyzing these cooling curves reveals a great deal of information about the structure and composition of neutron stars. However, model-observation comparisons of such cooling curves are challenging - partly due to observational uncertainties, and partly due to incomplete knowledge of heating mechanisms during accretion outbursts. This situation is further exacerbated by the recent discovery of Urca cooling in the neutron star crust. These are cycles that alternate between electron-capture and $\beta$-decay to produce a large flux of neutrinos and anti-neutrinos. These freely stream out of the star and carry energy with them, essentially cooling the neutron star crust without changing the composition. As a result, it is necessary to accurately quantify the strength of Urca cooling to constrain the heat sources in neutron star crusts and facilitate better model-observation comparisons of the cooling curves. Urca cooling is effective only for a certain subset of nuclei with specific properties. One of the required conditions is a strong ground-state to ground-state $\beta$-decay transition strength for a nucleus. Previous studies have shown $^{33}$Mg to be a strong Urca cooling agent for neutron star crusts composed of X-ray burst ashes. This is attributed to a 37\% strong ground-state branch in the $\beta$-decay of $^{33}$Mg inferred from high-resolution $\beta$-delayed $\gamma$ spectroscopy. This is, however, a first-forbidden transition and the strong ground-state branch seems anomalously high compared to theoretical calculations. A goal of this dissertation is to remeasure this transition strength using Total Absorption Spectroscopy with the SuN detector. $\beta$-delayed neutron branching ratio is also measured with the NERO detector. A combination of SuN and NERO helps mitigate the Pandemonium effect, which is shown to systematically overestimate low-energy branchings in high-resolution $\gamma$ spectroscopy. The ground-state branch for the $\beta$-decay of $^{33}$Mg $\rightarrow$ $^{33}$Al was measured to be 0.7(24)\% corresponding to a log-ft value of 7.0$_{-0.7}^{+\infty}$. This is significantly lower than the previous measurement and is consistent with the first-forbidden nature of the transition arising from the recently confirmed negative parity ground state of $^{33}$Mg. It further translates into a substantially reduced intrinsic Urca cooling luminosity of L$_\text{34}$ = 60.0. This highlights the importance of Total Absorption Spectroscopy and motivates future experiments with this technique to refine calculations of Urca cooling. The rate of Urca cooling (L$_{34}$) is extremely sensitive to electron-capture thresholds (Q$_\text{EC}$) and is proportional to |(Q$_\text{EC}$)|$^5$. These electron-capture thresholds depend on nuclear masses. Several of the potential Urca cooling candidates are neutron-rich exotic nuclei whose masses have not been measured experimentally and Urca cooling calculations have to rely on theoretical mass predictions. However, theoretical model predictions diverge as they move away from the stable nuclei and do not have uncertainties. A global nuclear mass model with quantified uncertainties is also developed as a part of this dissertation using Bayesian Gaussian Process Regression and Bayesian Model Averaging (BMA). Updated neutron star crust calculations with the BMA mass model change not only the magnitude of Urca cooling but also the depth at which it happens. This has important implications for the overall thermal profile of the accreting neutron star crust.
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