If we assumed that a black hole with a singularity of infinite density eventually evaporates due to Hawking Radiation, does this affect the singularity itself?
If mass is preserved in the singularity after crossing the event horizon (and partially converted to energy during absorption), and this mass is eventually emitted as Hawking Radiation: does this mean, the singularity evaporates? How can infinite density eventually exist with 0 mass? Infinitely small volume?
I believe it is fair to say no human being knows how to answer this question in a complete and fully reliable manner. In other words, you got to research level!
Black hole evaporation is a computation done using a framework known as quantum field theory in curved spacetime (QFTCS), which treats gravity as a classical phenomenon. This is most often understood as an approximation to a more fundamental theory of quantum gravity, which we currently do not know (although there are many candidates, such as String Theory, Loop Quantum Gravity, Asymptotically Safe Quantum Gravity, among many others). The thing about QFTCS is that while it is reliable for reasonably large scales (much larger than quantum gravity scales, which are about $10^{-35} \text{m}$, it is quite a stretch to use it for very small scales. The final stages of black hole evaporation would eventually get to these very small scales, and then the theory is simply not really that reliable, so we can't really say confidently what happens. We can argue why this or that might or not be possible, or why this or that might or not be paradoxical, but we don't really know what happens in those stages.
Moreover, we don't even know whether singularities are a physical thing or just an issue of classical gravity. It might as well be that singularities do not exist in the quantum theory. Many scientists believe quantum gravity will "cure" the singularities.
Summary:
Q: If we assumed that a black hole with a singularity of infinite density eventually evaporates due to Hawking Radiation, does this affect the singularity itself?
A: We do not know. In fact, we don't even know if the singularity is real or just a misdescription of General Relativity.
Q: does this mean, the singularity evaporates?
A: we do not know, since we don't even know if it exists.
Q: How can infinite density eventually exist with 0 mass?
A: The energy density of a singularity isn't really well defined in GR. Actually, the mass of a black hole is more of a global property, which you compute by means of expressions that resemble Gauss' law of Electromagnetism (for example, you can use the Komar formulae). Furthermore, we don't know if the singularity physically exists.
Q: Infinitely small volume?
A: Means quantum gravity. We have guesses at how it might be, but don't really know how it is.
