STEREO/EUVI observed 185 flare events (detected above the GOES class C1 level or at > 25 keV with RHESSI) during the first two years of the mission (December 2006 – November 2008), while coronal mass ejections (CMEs) were reported in about a third of these events. We compile a comprehensive catalog of these EUVI-observed events, containing the peak fluxes in soft X rays, hard X rays, and EUV, as well as a classification and statistics of prominent EUV features: 79% show impulsive EUV emission (coincident with hard X rays), 73% show delayed EUV emission from postflare loops and arcades, 24% represent occulted flares, 17% exhibit EUV dimming, 5% show loop oscillations or propagating waves, and at least 3% show erupting filaments. We analyze an example of each EUV feature by stereoscopic modeling of its 3D geometry. We find that EUV emission can be dominated by impulsive emission from a heated, highly sheared, noneruptive filament, in addition to the more common impulsive EUV emission from flare ribbons or the delayed postflare EUV emission that results from cooling of the soft-X-ray-emitting flare loops. Occulted flares allow us to determine CME-related coronal dimming uncontaminated from flare-related EUV emission. From modeling the time evolution of EUV dimming we can accurately quantify the initial expansion of CMEs and determine their masses. Further, we find evidence that coronal loop oscillations are excited by the rapid initial expansion of CMEs. These examples demonstrate that stereoscopic EUV data provide powerful new methods to model the 3D aspects in the hydrodynamics of flares and kinematics of CMEs.