Observational data are increasingly used to evaluate the effects of treatments on health outcomes. Causal inference provides a framework for formulating estimands of interest in this setting; however, identifiability of these estimands relies on certain assumptions. One assumption is called positivity, which requires the probability of treatment to be bounded away from 0 and 1. That is, for every covariate combination, we should observe both treated and control subjects. If the positivity assumption is violated, population-level causal inference necessarily involves some extrapolation. Ideally, a greater amount of uncertainty around the causal effect estimate is reflected in areas of non-overlap. With that goal in mind, we construct a Gaussian process model for estimating treatment effects in the presence of practical violations of positivity. Our method does not rely on strong parametric assumptions, provides a cohesive model for estimating treatment effects, and incorporates more uncertainty in areas of the covariate space where there is less overlap. Our work also focuses on the causal analysis of cluster randomized trials (CRTs) with a small number of clusters and a rare binary outcome. While estimation and covariate adjustment via generalized estimating equations (GEE) can account for chance imbalances and increase statistical power, analytical challenges frequently arise in such settings. For example, traditional GEE models tend to produce negatively biased standard error estimates, and regression adjustment often fails to converge with a rare outcome. We evaluate the utility of propensity score weighting and regression adjustment both in conjunction with bias-corrected sandwich variance estimators to precisely estimate a causal odds ratio and to obtain valid inference. In each project, we assess the proposed approaches and compare with alternative methods through simulation studies and then demonstrate their implementation with real use cases, including an observational study of right heart catheterization in female patients and a CRT that tests a sedation protocol in 31 pediatric intensive care units.