Is it possible to calculate the skin temperature of an object passing through a gas? If you have an object passing through a gas at a certain velocity, is it then possible to calculate the temperature of that object using properties like velocity, density, mass and/or others?
 A: Yes.
There are two effects: Compression heat and friction heat. Of course, convection, heat conductivity and radiation will add to that, like for any stationary object. But the movement through the gas adds the two mentioned effects on top.
Stagnation heat grows with the square of speed and affects the most forward part of the body. The formula for the stagnation point temperature $T_s$ of an ideal gas of the temperature $T_{\infty}$ hitting an object with the Mach number Ma is $$T_s = T_{\infty} + T_{\infty}\cdot\frac{(\kappa-1)\cdot Ma^2}{2}$$
For air the ratio of specific heats $\kappa$ is 1.4. Example: The tip of the fuselage nose of an airliner flying at Mach 0.85 will see air temperature rise by 14.45%. If the air at altitude has a temperature of 220 K (-53.15°C), the air temperature at the stagnation point will be 251.8 K (-21.36°C).
But past the stagnation point the air will accelerate and become faster than flight speed. Now pressure and, consequently, temperature need to drop sufficiently to encourage the flow to stay attached and follow the curvature of the forward fuselage. This acceleration will cool the air, so the flow right above the windshield will be cooler than ambient air.
Along the cylindrical portion of the fuselage, we find roughly flight speed again, but now friction will change the temperature close to the wall. Again the kinetic energy is converted, but the heating is caused by friction. See the boundary layer plots below:

Frictional and thermal boundary layer (picture source)
The temperature close to the wall is now called recovery temperature and is different from the stagnation point temperature because there is a small speed component normal to the surface which carries away some of the heat. The air temperature depends on the ratio between viscous diffusion and thermal diffusion, which is expressed by the Prandtl number Pr. If Pr>1, the air temperature at the wall is higher than the stagnation temperature and for Pr<1, it is colder. The Prandtl number of air is 0.72, so the air surrounding the fuselage is slightly colder than the stagnation temperature.
If you now expect a simple formula for the heat along the body: Sorry, no such luck. Details depend on the pressure distribution, the gas parameters and the temperature difference between gas and body. There is a special part of gas dynamics called Aerothermodynamics which covers this discipline.
