We investigate conditions for solvability and Malliavin differentiability of backward stochastic differential equations driven by a Lévy process. In particular, we are interested in generators which satisfy a local Lipschitz condition in the Z and U variable. This includes settings of linear, quadratic and exponential growths in those variables. Extending an idea of Cheridito and Nam to the jump setting and applying comparison theorems for Lévy-driven BSDEs, we show existence, uniqueness, boundedness and Malliavin differentiability of a solution. The pivotal assumption to obtain these results is a boundedness condition on the terminal value ξ and its Malliavin derivative Dξ. Furthermore, we extend existence and uniqueness theorems to cases where the generator is not even locally Lipschitz in U. BSDEs of the latter type find use in exponential utility maximization.