Chiral magnets support topological skyrmion textures due to the Dzyaloshinskii-Moriya spin-orbit interaction. In the presence of a sufficiently large applied magnetic field, such skyrmions are large-amplitude excitations of the field-polarized magnetic state. We investigate analytically the interaction between such a skyrmion excitation and its small-amplitude fluctuations, i.e., the magnons in a clean two-dimensional chiral magnet. The magnon spectrum is found to include two magnon-skyrmion bound states corresponding to a breathing mode and, for intermediate fields, a quadrupolar mode, which will give rise to subgap magnetic and electric resonances. Due to the skyrmion topology, the magnons scatter from an Aharonov-Bohm flux density that leads to skew and rainbow scattering, characterized by an asymmetric differential cross section with, in general, multiple peaks. As a consequence of the skew scattering, a finite density of skyrmions will generate a topological magnon Hall effect. Using the conservation law for the energy-momentum tensor, we demonstrate that the magnons also transfer momentum to the skyrmion. As a consequence, a magnon current leads to magnon pressure reflected in a momentum-transfer force in the Thiele equation of motion for the skyrmion. This force is reactive and governed by the scattering cross sections of the skyrmion; it causes not only a finite skyrmion velocity but also a large skyrmion Hall effect. Our results provide, in particular, the basis for a theory of skyrmion caloritronics for a dilute skyrmion gas in clean insulating chiral magnets.