Reflected or constrained diffusions in convex polyhedral domains arise in a variety of disciplines, including as approximations of queueing and chemical reaction networks, interacting particle systems, Leontief systems in economics, and in the study of Atlas models in math finance. We establish pathwise differentiability of a large class of obliquely reflected diffusions in convex polyhedral domains, and show that they can be described in terms of certain constrained stochastic differential equations with time-varying domains and directions of reflection. As an application, we use these derivatives to construct asymptotically unbiased estimators for expectations of functionals of reflected diffusions with respect to their defining parameters, such as the initial condition, drift and covariance coefficients, and directions of reflection. While differentiability of flows and sensitivity analysis are classical topics in dynamical systems, the analysis of these properties for constrained processes, which arise in a variety of applications, is challenging due to the discontinuous dynamics at the boundary of the domain, and is further complicated when the boundary is non-smooth. A key step in our study of both differentiability of flows and sensitivities of constrained diffusions in convex polyhedral domains is the study of related directional derivatives of an associated map, called the Skorokhod map, which may be of independent interest.