We introduce notions of finite presentation which serve as analogues of finite-dimensionality for operator modules over completely contractive Banach algebras. With these notions we then introduce analogues of the local lifting property, nuclearity, and semi-discreteness. For a large class of operator modules, we show that the local lifting property is equivalent to flatness, generalizing the operator space result of Kye and Ruan. We pursue applications to abstract harmonic analysis, where, for a locally compact group G, we show that A(G)-nuclearity of the inclusion $C*_r(G) \to C^*_r(G)**$ and $A(G)$-semi-discreteness of $VN(G)$ are both equivalent to amenability of $G$. We also present the equivalence between $A(G)$-injectivity of the crossed product $G\bar{\ltimes}M$, $A(G)$-semi-discreteness of $G\bar{\ltimes} M$, and amenability of W*-dynamical systems $(M,G,\alpha)$ with $M$ injective.