Is this analogy of Hawking Radiation correct?
No. I know you read stuff like this on the internet, in pop-science articles, and even in textbooks. But it's not correct at all. Sorry. I'll go through it step by step:
Within the ergosphere of the black hole, virtual pairs of particles and anti-particles are constantly appearing
No they aren't. Virtual particles are virtual. They aren't short-lived real particles. See Do virtual particles actually physically exist? and note this in the answer by retired particle physicist anna v:
"virtual particles exist only in the mathematics of the model".
Also see Matt Strassler's article. (Ah I see John Rennie referred to it too). The point is that electrons and protons don't throw photons at one another. Virtual photons aren't short-lived real photons popping in and out of existence. Instead they're "field quanta". It's like you divide an electromagnetic field into abstract portions, and say each is a virtual photon. Then when the electron and the proton attract one another, they exchange field, such that the resultant hydrogen atom doesn't have much of an electromagnetic field left. Hence you can see the underlying correctness of the exchange idea. But electrons don't flash, hydrogen atoms don't twinkle, and magnets don't shine.
due to vacuum fluctuations.
Vacuum fluctuations aren't the same thing as virtual particles. Instead they're the spatial equivalent of the tiny little ripples you see on the surface of the sea. They are real, but they're very weak. See the Casimir effect on Wikipedia:
CCASA image by Emok, see Wikipedia
Typically, the pair would then annihilate before this could be of any consequence.
Vacuum fluctuations don't result in pair production. And electrons and positrons don't just pop into existence like magic, and then obligingly annihilate without leaving two 511keV photons.
However, as one of these two particles will be closer to the hole than the other, it will experience a greater gravitational force than the other. Thus, there is the possibility that one particle will fall into the black hole while the other escapes to infinity.
If this tale was true, it would be more likely that both particles would fall into the hole, which would grow more massive as a result. It would in effect be feeding on vacuum energy.
The particle that falls in will have negative energy, and subsequently negative mass, leaving the positive energy particle outside of the hole.
Do you know of any particles with negative energy? Or negative mass? If you do, please tell me, I'll write about it, and then I'll wait for the call from Stockholm. There are no such particles.
It is not the black hole that is directly emitting the particle, but from an external reference frame, this appears to be case. As tidal forces are greater for smaller black holes, the rate of emission of Hawking radiation increases as the hole decreases in size. In an isolated system, this would lead to an exponential decay. However, black holes are constantly accreting mass too. As such, any black hole with a sufficient mass will accrete mass faster than it loses it via Hawking radiation. Yet this is not the case with smaller holes, which would slowly 'evaporate'.
That's what people say. But think about gravitational time dilation. It goes infinite at the event horizon. Why is there no mention of this in the given explanation of Hawking radiation?
As this is all outside of my school curriculum and way above my current level of taught physics, I was hoping someone could either confirm that my analogy is correct or provide some constructive criticism and advice. Thanks!
It's a fairy story I'm afraid. Hawking radiation remains hypothetical even after 43 years. But it's been around so long that people believe in it. People take it as scientific fact, even though there's no scientific evidence for it, even though it doesn't square with general relativity, and even though the given explanation is just "lies to children". See this little physicsworld article for a hint of what some physicists think of Hawking. IMHO you should focus on gravity and black holes and avoid getting distracted by Hawking radiation.