We investigate the conformation, position, and dynamics of core-shell nanoparticles (CSNPs) composed of a silica core encapsulated in a cross-linked poly(N-isopropylacrylamide) shell at a water-oil interface for a systematic range of core sizes and shell thicknesses. We first present a free-energy model that we use to predict the CSNP wetting behavior at the interface as a function of its geometrical and compositional properties in the bulk phases, which is in good agreement with our experimental data. Remarkably, based on the knowledge of the polymer shell deformability, the equilibrium particle position relative to the interface plane, an often elusive experimental quantity, can be extracted by Shell thickness measuring its radial dimensions after adsorption. For all the systems studied here, the interfacial dimensions are always larger than in bulk and the particle core resides in a configuration, wherein it just touches the interface or is fully immersed in water. Moreover, the stretched shell induces a larger viscous drag at the interface, which appears to depend solely on the interfacial dimensions, irrespective of the portion of the CSNP surface exposed to the two fluids. Our findings indicate that tailoring the architecture of CSNPs can be used to control their properties at the interface, as of interest for applications including emulsion stabilization and nanopatterning.