2 different density liquids in deep space
The question whether two liquids mix or not is mainly to do with the intermolecular forces, not the densities. These forces lead to the bulk property known as chemical potential.
Assuming the experiment is done in a pressurized cabin, then the ambient temperature and pressure may take ordinary values such as STP (room temperature, 1 atmosphere of pressure).
At such ordinary temperature and pressure, what happens with oil and water is that the fluid consists of two parts, one of which is mostly (but not completely) water and the other is mostly (but not completely) oil. The entropy of system-plus-environment is maximised when the Gibbs function is minimised, with the result that the oil is not fully dissolved but rather the liquid consists of these two parts. The two parts will gather at separate locations so as to minimise any boundary areas where there is surface tension, and also in response to any ambient forces such as gravity.
In the absence of gravity, therefore, one expects a blob of one liquid and a blob of the other. There are three types of boundary: water-oil, water-air, oil-air, each with a different surface tension. I think the water-air boundary has the highest surface tension, which suggests the oil will surround the water so as to elliminate this boundary. Therefore if there is enough oil then I expect the equilibrium configuration is a sphere of water surrounded by a spherical shell of oil. At smaller amounts of oil I guess the oil will be spread out over the surface of the water, not quite covering it.
Perhaps the configuration with the least Gibbs potential is some other shape (depending on the proportions of oil and water), but the equilibrium configuration will certainly consist of two parts with different concentrations as I have said, unless the temperature is high. At high temperature the two fluids fully dissolve into one another and then you would have a single continuous substance, not two.
With no gravity and no impulse given to the blobs of fluid, my guess is that they would remain one inside the other and do nothing. If present, gravity would play on the blobs of fluid by making the less dense fluid (the oil) rise relative to the denser fluid, due to buoyancy. The oil would also "rise" if the spacecraft were rotating due to the greater inertia of the denser water. If your oil blob was injected off centre, or the surrounding shell of water was thin such that the minimum distance between the oil-water and water-air interfaces is on the scale of the capillary length, something funky might happen. My guess is that the water blob and oil blob might change places as the water (which has a higher surface tension than the oil) tries to minimise its surface energy by assuming the smallest size possible: a spherical blob of radius $r$ would have a smaller surface energy than a shell of inner and outer radii $r$ and $r+\delta r$, respectively.