This paper provides two generalizations of differential and integral categories: Leibniz and generalized Rota-Baxter categories, which capture certain algebraic structures, and q-categories, which capture structures of quantum calculus. In the search for new examples of differential and integral categories, it was observed that many structures were not quite examples but satisfied certain properties and not others. This leads us to the definition of Leibniz, Rota-Baxter and proto-FTC categories. In generalizing Rota-Baxter categories further to an arbitrary weight, we show that we recapture Ribenboim's generalized power series as a monad on vector spaces with a generalized integral transformation. This also subsumes the renormalization operator on Laurent series, which has applications in the quantum realm. Finally, we define quantum differential and quantum integral categories, show that they recapture the usual notions of quantum calculus on polynomials, and construct a new example to indicate their potential usefulness outside of that specific setting