We investigate the X-ray and near-infrared emission properties of a sample of pre-main-sequence (PMS) stellar systems in the Orion Nebula Cluster (ONC) that display evidence for circumstellar disks ("proplyds") and optical jets in Hubble Space Telescope (HST) imaging. Our study uses X-ray data acquired during Chandra Orion Ultradeep Program (COUP) observations, as well as complementary optical and near-infrared data recently acquired with HST and the Very Large Telescope (VLT), respectively. Approximately 70% of ~140 proplyds were detected as X-ray sources in the 838 ks COUP observation of the ONC, including ~25% of proplyds that do not display central stars in HST imaging. In near-infrared imaging, the detection rate of proplyd central stars is >90%. Many proplyds display near-infrared excesses, suggesting disk accretion is ongoing onto the central, PMS stars. About 50% of circumstellar disks that are detected in absorption in HST imaging contain X-ray sources. For these sources, we find that X-ray absorbing column and apparent disk inclination are well correlated, providing insight into the disk scale heights and metal abundances of UV- and X-ray-irradiated protoplanetary disks. Approximately 2/3 of the ~30 proplyds and PMS stars exhibiting jets in Hubble images have COUP X-ray counterparts. These jet sources display some of the largest near-infrared excesses among the proplyds, suggesting that the origin of the jets is closely related to ongoing, PMS stellar accretion. One morphologically complex jet source, d181-825, displays a double-peaked X-ray spectral energy distribution with a prominent soft component that is indicative of strong shocks in the jet collimation region. A handful of similar objects also display X-ray spectra that are suggestive of shocks near the jet source. These results support models in which circumstellar disks collimate and/or launch jets from young stellar objects and, furthermore, demonstrate that star-disk-jet interactions may contribute to PMS X-ray emission.