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A supersonic relative velocity between dark matter (DM) and baryons (the stream velocity) at the time of recombination induces the formation of low mass objects with anomalous properties in the early Universe. We widen the scope of the `Supersonic Project' paper series to include objects we term Dark Matter + Gas Halos Offset by Streaming (DM GHOSts)--diffuse, DM-enriched structures formed because of a physical offset between the centers of mass of DM and baryonic overdensities. We present an updated numerical investigation of DM GHOSts and Supersonically Induced Gas Objects (SIGOs), including the effects of molecular cooling, in high resolution hydrodynamic simulations using the AREPO code. Supplemented by an analytical understanding of their ellipsoidal gravitational potentials, we study the population-level properties of these objects, characterizing their morphology, spin, radial mass, and velocity distributions in comparison to classical structures in non-streaming regions. The stream velocity causes deviations from sphericity in both the gas and DM components and lends greater rotational support to the gas. Low mass ($<\sim 10^{5.5}$ M$_\odot$) objects in regions of streaming demonstrate core-like rotation and mass profiles. Anomalies in the rotation and morphology of DM GHOSts could represent an early Universe analogue to observed ultra-faint dwarf galaxies with variations in DM content and unusual rotation curves.