Hawking Radiation

I understand Hawking Radiation occurs when two particles appear at the event horizon of a black hole, but instead of annihilating each other and "returning" the energy to the outside, the negatively-charged particle is sucked into the black hole and the positively charged particle is "released" by the black hole, appearing to an outside observer as though the black hole emitted some energy.

I'd like to know: what happens if the positively-charged particle is sucked into the black hole and the negatively-charged particle is released? Wouldn't this increase the black hole's mass? If so, why do black holes lose mass over time? Shouldn't they lose some, then gain some, and so on, thus never completely losing all their mass?

And as a side question: why do the two particles appear to begin with? My current understanding is that it has to do with Heisenberg's Uncertainty Principle, and the lack of matter at the event horizon: you'd be able to know the amount of energy with absolute certainty (0) in any given time frame, so particles must appear, seemingly randomly, to maintain the principle in balance, like the opposite of what happens with Zero-Point Energy (and yet the same thing, in a way). Is this correct?

Thank you in advance.