The kinetic Sunyaev-Zel'dovich (kSZ) effect results from Thomson scattering by coherent flows in the reionized intergalactic medium. We present new results based on ray-tracing an 8 Gpc/h realization of reionization with resolution elements 2 Mpc/h (subtending similar to 1' at z = 6) on a side to create a full-sky kSZ map. The realization includes, self-consistently, the effects of reionization on scales corresponding to multipoles 10 less than or similar to l less than or similar to 5000. We separate the kSZ map into Doppler (v), Ostriker-Vishniac (delta v), patchy (xv), and third-order (x delta v) components, and compute explicitly all the auto-and cross-correlations (e.g., < vv >, <delta vxv >, etc.) that contribute to the total power. We find a complex and nonmonotonic dependence on the duration of reionization at l similar to 300 and evidence for a non-egligible (10%-30%) contribution from connected four-point correlations, < xvxv >(c), usually neglected in analytical models. We also investigate the cross-correlation of linear matter and large-scale kSZ temperature fluctuations, focusing on (1) cross-power spectra with biased tracers of the matter density and (2) cold spots from infall onto large, rare H II regions centered on peaks in the matter distribution at redshifts z > 10 that are a generic non-Gaussian feature of patchy reionization. Finally, we show that the reionization history can be reconstructed at 5 sigma-10 sigma significance by correlating full-sky 21 cm maps stacked in bins with Delta(nu) = 10 MHz with existing cosmic microwave background (CMB) temperature maps at l < 500, raising the prospects for probing reionization by correlating CMB and LSS measurements. The resulting kSZ maps have been made publicly available at www.cita.utoronto.ca/similar to malvarez/research/ksz-data/.