Accreting protoplanets are windows into planet formation processes, and high-contrast differential imaging is an effective way to identify them. We report results from the Giant Accreting Protoplanet Survey (GAPlanetS), which collected Ha differential imagery of 14 transitional disk host stars with the Magellan Adaptive Optics System. To address the twin challenges of morphological complexity and point-spread function instability, GAPlanetS required novel approaches for frame selection and optimization of the Karhounen-Loeve Image Processing algorithm pyKLIP. We detect one new candidate, CS Cha "c," at a separation of 68 mas and a modest Delta mag of 2.3. We recover the HD 142527 B and HD 100453 B accreting stellar companions in several epochs, and the protoplanet PDS 70 c in 2017 imagery, extending its astrometric record by nine months. Though we cannot rule out scattered light structure, we also recover LkCa 15 "b," at Ha; its presence inside the disk cavity, absence in Continuum imagery, and consistency with a forward-modeled point source suggest that it remains a viable protoplanet candidate. Through targeted optimization, we tentatively recover PDS 70 c at two additional epochs and PDS 70 b in one epoch. Despite numerous previously reported companion candidates around GAplanetS targets, we recover no additional point sources. Our moderate Ha contrasts do not preclude most protoplanets, and we report limiting Ha contrasts at unrecovered candidate locations. We find an overall detection rate of similar to 36 -22+26%