When is it a good aproximation to consider a star to be an ideal gas?
There is no hard and fast answer. To be treated as an idea gas (your title question), the particles in your gas should be point-like and they should be non-interacting if you are to use the ideal gas approximation.

This means you can take the mean separation ($\sim n^{-1/3}$) and compare that with the size of the particles - it should be much bigger. You can also compare the particle kinetic energies with the potential energy of their interactions at typical separations. eg The ratio of $kT$ to the Coulomb potential energy should be large.

The rest of your question is confused. You are asking about an adiabatic index, but that is a completely separate question from whether the gas is ideal. It depends on the equation of state, on interactions, on internal degrees of freedom, on the way energy is transported.

Relativistically degenerate gases can be ideal gases and are often assumed to be so. Do not confuse an ideal gas with a perfect gas (even though many do so).