By performing molecular dynamics simulations, we investigate the structural and dynamical properties of polymer melts containing probe spherical nanoparticles. Generally speaking, the behavior of these polymer nanocomposites is strongly affected by the interaction strength established between the nanoparticles and the chain monomers and by the nanoparticle sizes. We highlight that this dependence is not always evident and some intriguing properties, such as the heterogeneous dynamics of both polymer chains and nanoparticles and their nonGaussian behavior at short and long timescales, are not particularly influenced by the degree of attraction between nanoparticles and polymer for the range of interactions we study (up to 6 k(B)T). We find the existence of weakly ordered interdigitated structures with sequential arrangements of particles and polymer chains, which separate each other and hence inhibit the formation of nanoparticle clusters. This is especially evident with big nanoparticles, being less prone to aggregate than small ones, even when their interaction with the polymer chain is as low as 0.5 k(B)T. Moreover, by integrating the stress-tensor autocorrelation functions, we estimate the shear viscosity and determine its dependence on the strength of the polymer-nanoparticle interactions and on the nanoparticle size. By acting as plasticizers, small nanoparticles decrease the viscosity, especially at low-to-moderate interactions with the polymer. By contrast, big nanoparticles that establish strongly attractive interactions with the polymer chains behave as thickening agents and significantly increase the viscosity. This complex and perhaps still scantily understood balance between the geometry of nanoparticles and their interaction with the polymer is key to predict and fully control the macroscopic response of nanocomposite materials and hence suitably tailor their mechanical properties.