Hessenberg differential-algebraic equations (DAEs) of size ν with nonsmooth right-hand side functions are studied. Well-posedness theory established in this article includes consistent initialization robust to parametric perturbations, and local existence and uniqueness of solutions in the classical senses. The aforementioned results follow from “regularity” of initial data, which is stated in terms of participating functions determining a Lipschitz homeomorphism. Said regularity implies the nonsmooth Hessenberg DAEs have (local) generalized differentiation index ν, and can be verified with matrix-theoretic conditions involving, for example, Clarke Jacobian projections. Regular solutions on an interval of interest are shown to have maximal continuations and to be lexicographically smooth with respect to problem parameters. Consequently, lexicographic directional differentiation can be applied to yield an auxiliary, nonsmooth “high-index” sensitivity system whose solution characterizes generalized derivative information of the reference solution of interest with respect to parameters. This theory is computationally relevant as the parametric sensitivity functions can be evaluated in a tractable way due to recent progress in nonsmooth automatic/algorithmic differentiation, and can be supplied to dedicated nonsmooth numerical equation-solving or optimization methods. This allows for numerically implementable dynamic optimization methods; open-loop optimal control theory is given which provides subdifferential elements of objective functions via solving a nonsmooth equation system involving the parametric sensitivity functions. As a corollary of the results in this article, foundational theory for smooth Hessenberg DAEs having classical differentiation index ν is established in a rigorous manner.