We study the properties of powerful X-ray flares from 161 pre-main-sequence (PMS) stars observed with the Chandra X-ray Observatory in the Orion Nebula region. Relationships between flare properties, protoplanetary disks, and accretion are examined in detail to test models of star-disk interactions at the inner edge of the accretion disks. Previous studies found no differences in flaring between disk-free and accreting systems other than a small overall diminution of X-ray luminosity in accreting systems. The most important finding is that X-ray coronal extents in fast-rotating disk-free stars can significantly exceed the Keplerian corotation radius, whereas X-ray loop sizes in disky and accreting systems do not exceed the corotation radius. This is consistent with models of star-disk magnetic interaction in which the inner disk truncates and confines the PMS stellar magnetosphere. We also find two differences between flares in accreting and disk-free PMS stars. First, a subclass of superhot flares with peak plasma temperatures exceeding 100 MK are preferentially present in accreting systems. Second, we tentatively find that accreting stars produce flares with shorter durations. Both results may be consequences of the distortion and destabilization of the stellar magnetosphere by the interacting disk. Finally, we find no evidence that any flare types, even slow-rise top-flat flares, are produced in star-disk magnetic loops. All are consistent with enhanced solar long-duration events with both footpoints anchored in the stellar surface.