  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.
