How long does it take (in terms of years) for energy (a photon) to get from the core to the surface of a white dwarf with effective temperature of 30,000K and average inner temperature of 15MK ? I think the formula we should employ is: $$t = (3R^2) / (lc)$$ where $R$ is the radius of the star, $l$ is the mean free path and $c$ is the speed of the photon. I can't figure out how we can calculate $R$ and $l$ based on the information given in the question.
You can't calculate $R$ and $l$ from the information given. $R$, the radius of the star, has nothing to do with electron scattering through it---i.e. it's an external parameter that you have to know or look up (White Dwarf radius is a good value to remember --- it's basically the same as the Earth!). The mean free path you determine by the properties of the medium through which you are scattering.
If this is Thomson scattering of photons with electrons, then how can you calculate the mean free path? If you consider things like mass, number density, charge, "cross section", velocity... what needs to go into the equation?
Please comment if you need further help / hints