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Checking for electron degeneracy is a matter of comparing the Fermi kinetic energy with $kT$. If $E_F/kT \gg 1$, then you may assume the electrons are degenerate. The central density of the Sun is around $\rho=1.6\times 10^5$ kg/m$^3$ and the number of atomic mass units per electron is around $\mu_e =1.5$. The number density of electrons is therefore ...

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The star acts as an "effective point source" since it is so far away and its angular extent is so small - in other words, the optical signal arriving at earth is "very nearly" a plane wave with the same phase over a large extent. This is what enables us to do interferometry. Think of waves, not photons.

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You should just think of Deuterium burning as a short-lived counterpart of main sequence hydrogen burning. As the pre-main-sequence star (PMS star, or protostar if you like) contracts, its core reaches around $6\times 10^{5}$ K and D burning is smoothly initiated. The star is then held at roughly constant luminosity and radius, because if it contracted, the ...

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