We present a comprehensive investigation of the size-dependent switching characteristics and spin wave modes of FePt nanoelements. Curved nanomagnets ("caps") are compared to flat disks of identical diameter and volume over a size range of 100 to 300nm. Quasi-static magnetization reversal analysis using first-order reversal curves (FORC) shows that spherical caps have lower vortex nucleation and annihilation fields than the flat disks. As the element diameter decreases, the reversal mechanism in the caps crosses over sooner to coherent rotation than in the disks. The magnetization dynamics are studied using optically induced small angle precession and reveal a strong size dependence that differs for the two shapes. Flat disks exhibit well-known center and edge modes at all sizes, but as the diameter of the caps increases from 100 to 300 nm, additional oscillation modes appear in agreement with dynamic micromagnetic simulations. In addition, we show that the three-dimensional curvature of the cap causes a much greater sensitivity to the applied field angle which provides an additional way for controlling the ultrafast response of nanomagnetic elements.