Understanding causality from observational data is often of interest in scientific research. We propose methods for identifying and estimating causal parameters related to treatment effect heterogeneity and sensitivity analysis. In the first project, we study two causal parameters defined as functions of the conditional average treatment effect (CATE): the average squared CATE and the variance of CATE. These parameters measure the overall extent of treatment effect and treatment effect heterogeneity across the population, respectively. We develop efficient estimators for these parameters, which are guaranteed to be in the parameter space and allow for valid inference under both the null and alternative hypotheses. We demonstrate the practical performance of our methodology through numerical studies and a real data application. In the second and third projects, motivated by assessing evidence of causal effects of physical activity on mortality given potential unobserved confounding or confounder misclassification, we study sensitivity analyses to unobserved confounding and confounder misclassification with time-to-event outcomes. In the second project, we compare two types of causal sensitivity analyses using data from the NIH-AARP Study: to confounder misclassification, and to unobserved confounding. We find that the effect of physical activity on respiratory disease mortality is not explained away by a moderate amount of unobserved confounding or confounder misclassification. The effect of physical activity on cancer mortality is explained away by a small amount of unobserved confounding, but not by confounder misclassification. We hypothesize that the robustness to confounder misclassification could be due to assumptions of the measurement error model. These results indicate that existing sensitivity analysis tools require strong assumptions on the data generating process, making them not accurate enough to capture confounding structures in real-world scenarios. To tackle this limitation, in the third project, we extend a nonparametric sensitivity analysis framework (Chernozhukov et al., 2022) to time-to-event data. We propose efficient estimators to bound the causal effect as a function of unobserved confounding and demonstrate the performance of our proposed methods using numerical studies. We illustrate the use of proposed sensitivity analysis with data from the same prospective cohort study in the second project.