temperature - What is the minimum size of a ball of gas to become a star?

There is no equation, you need a detailed model of the interior physics of very low mass stars. Very roughly, you can say that
hydrogen fusion occurs when the central temperature exceeds about $10^{7} K$ (the density dependence is secondary) and that from the virial theorem, the central temperature is given approximately by $T simeq 1.6times 10^{7} M/R$, where the mass and radius are in solar units. $M/R$ decreases slowly down the main sequence towards lower mass stars, but then what happens is that electron degeneracy pressure becomes important and less massive objects do not become much smaller than Jupiter-sized and thus $M/R$ decreases and fusion stops - or rather it never starts.

In detail the minimum mass for hydrogen fusion in a manner that is capable of sustaining a star in equilibrium against gravitational contraction is about $0.075M_{odot}$. With an uncertainty of about $0.002M_{odot}$.

It is slightly more complicated than this, since at lower internal temperatures the deuterium in a star can fuse. This will happen during the early life of any ball of gas more massive than about 13 Jupiter masses.

Your assumptions about the relative H and He abundances are completely wrong. Even in the early universe the gas is $sim 25$% He by mass.