How does a liquid melt a hotter solid? I don't have a degree in anything, but I do know basic rules of science I believe. It seems pretty intuitive that heat will always flow from something hotter into something colder in order to reach an equilibrium.

My understanding of temperature is that it is a measure of how fast something's particles are vibrating. 

I've been very interested in the mechanics of salt water and ice spontaneously dropping in temperature because it seems counter intuitive and goes against everything I seem to know.

As far as I know, pure ice begins to melt at 0°C and above. If I were to place an ice cube at -2°C in oil that is -5°C, it wouldn't melt because the oil has less energy than the ice cube. If anything, energy would leave the ice cube to warm the oil and the cube of ice would get colder.

So, if I have salt water at say -2°C, and I put an ice cube in the mix that is pure ice and is 0°C, how is that colder water going to impart energy into something that's warmer (the ice cube). This seems extremely bizarre and seems to go against basic rules of thermodynamics.

I know that adding more salt would lower the melting point even further, but isn't a temperature a temperature? Does the material matter? Do two substances with the same temperature contain the same kinetic energy on the molecular level? If not, then what is temperature even a measurement of?

Thank you.
The cold salt water is going to take heat out of the ice cube just as you say, (and so the salt water will be heated a little to an extent depending on their relative amounts  etc.) The ice surface in contact with the salt water will dissolve as the freezing point of the salt water is lower than that of the ice. Eventually all the ice  will dissolve, but this will have taken some small amount of energy to do as explained next. (I assume little ice and lots of solution but whether all the ice will in fact melt will depend on the exact composition and could be found from a temperature vs. mole fraction plot for a solid liquid equilibrium at constant pressure. There could be solid salt-ice &amp; ice, solution &amp; ice, solution &amp; solid salt-ice or just solution.)

Dissolving a solid in a liquid, say salt in water, can cause a drop in temperature (provided the container is insulated) because it takes  energy to dissolve compounds and this can only come from the liquid. (The ions or molecules in the solid must be attracted to one another, by ionic or van der Waals forces, and so to be separated takes energy). The measure is called the heat of fusion (or melting), or enthalpy of fusion which is a positive quantity, for common salt it is $\approx 30 \pu{~kJ mol^{-1}}$ and of ice $\approx 6 \pu{~kJ mol^{-1}}$.