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4

The R in that equation is the distance from the star to observer, not the star radius. The light emitted from the star is distributed uniformly on a sphere of radius R, and when the light arrives to the Earth, that sphere will have a radius equal to the distance Earth-star. Therefore, the second relation for the two fluxes is about the apparent magnitudes (...

4

The short answer is that the redshift of stars that are close enough to put into H-R diagrams (or "color magnitude diagrams", if we're being precise, given that you're talking about using colors) is so small that the effects are minimal. The correction to color you're thinking of (called "K correction") depends on the redshift: the ...

2

Yes it would. It is that way because the effective temperature is defined to be $(L/4\pi \sigma R^2)^{0.25}$. The radius of a neutron star is about 10 km $(1.4\times 10^{-5}R_\odot)$. They are born with surface temperatures of around $10^8$ K. The coldest white dwarfs have effective temperatures of about 3000 K. The luminosity ratio is  \frac{L_{\rm NS}}{...

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