We combine polymer integral equation theory and computational chemistry methods to study the interfacial structure, effective interactions, miscibility and spatial dispersion mechanism of fullerenes dissolved in specific random AB copolymer melts characterized by strong noncovalent electron donor-acceptor interactions with the nanofiller. A statistical mechanical basis is developed for designing random copolymers to optimize fullerene dispersion at intermediate copolymer compositions. Pair correlation calculations reveal a strong sensitivity of interfacial packing near the fullerene to copolymer composition and adsorption energy mismatch. The potential of mean force between fullerenes displays rich trends, often nonmonotonic with copolymer composition, reflecting a nonadditive competition between direct filler attractions and polymer-mediated bridging and steric stabilization. The spinodal phase diagrams are in qualitative agreement with recent solubility limit experimental observations on three systems, and testable predictions are made for other random copolymers. The distinctive nonmonotonic variation of miscibility with copolymer composition is found to be primarily a consequence of composition-dependent, spatially short-range attractions between the A and B monomers with the fullerene and nontrivial pair correlations. A remarkably rich, polymer-specific temperature dependence of the spinodal diagram is predicted, which reflects the thermal sensitivity of spatial correlations which can result in fullerene miscibility either increasing or decreasing with cooling. The calculations are contrasted with a simpler effective homopolymer model and the random structure Flory-Huggins model. The former appears to be qualitatively reasonable but can incur large quantitative errors since it misses preferential packing of monomers near nanopartides, while the latter appears to fail qualitatively due to its neglect of all spatial correlations.