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We present near- and mid-infrared (IR) broadband imaging observations of the edge-on protoplanetary disk around HH 30 with the James Webb Space Telescope/Near Infrared Camera (NIRCam) and the Mid-Infrared Instrument (MIRI). We combine these observations with archival optical/near-IR scattered light images obtained with the Hubble Space Telescope (HST) and a millimeter-wavelength dust continuum image obtained with the Atacama Large Millimeter/submillimeter Array (ALMA) with the highest spatial resolution ever obtained for this target. Our multiwavelength images clearly reveal the vertical and radial segregation of micron-sized and sub-mm-sized grains in the disk. In the near- and mid-IR, the images capture not only bi-reflection nebulae separated by a dark lane but also diverse dynamical processes occurring in the HH 30 disk, such as spiral- and tail-like structures, a conical outflow, and a collimated jet. In contrast, the ALMA image reveals a flat dust disk in the disk midplane. By performing radiative transfer simulations, we show that grains of about 3 $\mu$m in radius or larger are fully vertically mixed to explain the observed mid-IR scattered light flux and its morphology, whereas millimeter-sized grains are settled into a layer with a scale height of $\gtrsim1$ au at $100$ au from the central star. We also find a tension in the disk inclination angle inferred from optical/near-IR and mm observations with the latter being closer to an exactly edge-on. Finally, we report the first detection of the proper motion of an emission knot associated with the mid-IR collimated jet detected by combining two epochs of our MIRI 12.8-$\mu$m observations.