That other question mentioned in the question-comments also discussed annihilation of particles and neutralization of electric charge inside the event horizon. In either question, the no-hair theorem trumps all. If GR is the end of the story, particle identity is destroyed by the singularity. Even if post-GR theories of gravity rescue the Universe from the creation of singularities, it doesn't matter because the form of the mass-energy inside the event horizon doesn't matter to the outside world.
The phrase black hole tends to be used without specifying exactly what it means, and defining exactly what you mean is important to answer your question.
The archetypal black hole is a mathematical object discovered by Karl Schwarzschild in 1915 - the Schwarzschild metric. The curious thing about this object is that it contains no matter. Techically it is a vacuum solution to Einstein's equations. There is a parameter in the Schwarzschild metric that looks like a mass, but this is actually the ADM mass i.e. it is a mass associated with the overall geometry. I suspect this is what you are referring to in your second paragraph.
The other important fact you need to know about the Schwarzschild metric is that it is time independent i.e. it describes an object that doesn't change with time and therefore must have existed for an infinite time in the past and continue to exist for an infinite time into the future. Given all this you would be forgiven for wondering why we bother with such an obviously unrealistic object. The answer is that we expect the Schwarzschild metric to be a good approximation to a real black hole, that is a collapsing star will rapidly form something that is in practice indistinguishable from a Schwarzschild black hole - actually it would form a Kerr black hole since all stars (probably) rotate.
To describe a real star collapsing you need a different metric. This turns out to be fiendishly complicated, though there is a simplified model called the Oppenheimer-Snyder metric. Although the OS metric is unrealistically simplified we expect that it describes the main features of black hole formation, and for our purposes the two key points are:
the singularity takes an infinite coordinate time to form
the OS metric can't describe what happens at the singularity
Regarding point (1): time is a complicated thing in relativity. Someone watching the collapse from a safe distance experiences a different time from someone on the surface of the collapsing star and falling with it. For the outside observer the collapse slows as it approaches the formation of a black hole and the black hole never forms. That is, it takes an infinite time to form the black hole.
This isn't the case for an observer falling in with the star. They see the singularity form in a finite (short!) time, but ... the Oppenheimer-Snyder metric becomes singular at the singularity, and that means it cannot describe what happens there. So we cannot tell what happens to the matter at the centre of the black hole. This isn't just because the OS metric is a simplified model, we expect that even the most sophisticated description of a collapse will have the same problem. The whole point of a singularity is that our equations become singular there and cannot describe what happens.
All this means that there is no answer to your question, but hopefully I've given you a better idea of the physics involved. In particular matter doesn't mysteriously cease to exist in some magical way as a black hole forms.
Best Answer
You ask how atoms can be that tightly compressed. Atoms are made of electrons and quarks (the protons and neutrons are made of quarks) and as far as we know electrons and quarks are point like i.e. they have no size. So in principle they can be compressed to infinite density if you squeeze hard enough. At this point someone is going to point out that all particles are ultimately made from strings, and assuming string theory is correct we do expect the rules to change at sizes of around the string scale. However this is probably about $10^{-34}$ of a metre so let's ignore it for now.
Anyhow, quarks and electrons resist being compressed together for various reasons, so under everyday conditions we see everyday densities. The thing is that gravity is always additive - more mass means more gravity and more pressure, and we can keep piling on more matter and the gravity and pressure will keep rising. At some point the pressure gets so great that electrons react with proton to form neutrons, and we get matter made up just from neutrons. This is called neutronium and it's the state of matter found in neutron stars. Neutronium has a density of around $10^{18}$ kg/m$^3$ while ordinary matter is between $10^{3}$ and $10^{4}$ kg/m$^3$
When you try to compress neutrons they resist due to a phenomenon called degeneracy pressure, but you can keep adding mass and this keeps increasing the pressure until even the degeneracy pressure can't keep the neutrons apart. At that point it's unclear exactly what happens because we don't understand the physics that well. However it's possible that the next stage is that the neutrons dissolve into a sea of quarks and it may form something like strange matter, which has it's own degeneracy pressure that resists further compression.
But if you add even more mass you can overcome even the quark degenracy pressure and at that point the quarks start to collapse. Remember that I started out saying quarks are point like, so when they start to collapse they can collapse without limit and the density will become infinite.
That in a nutshell is how you can collapse the Earth to the size of a pea.
Well, maybe not. If string theory is correct quarks aren't point like particles. Once you go down to length scales around the Planck length the quark would start behaving like a string not a point particle, and we expect the rules to change. No-one knows what will happen, but it seems likely the collapse would be stopped before the density becomes infinite.