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I have a few more questions on black holes.

I have read that black holes have an infinite density, but I have also read that one of the properties of the black hole is its size. By size do they mean the size of the event horizon? Because something with infintie density becomes a point, right? If the density and curvature of spacetime is infinite, then does time stop, like it would if someone were to move at light speed, according to someone outside the event horizon?

And finally, for black holes evaporating, does the quantum process have to happen outside the event horizon? It seems like it has to. The book I'm reading is Stephen Hawkings A Brief History of Time, which may be too brief, kinda went to fast over the black holes evaporating. So when the pair of virtual particles is produced, does the process need to happen really close to the black holes event horizon, so that one of the particles will fall into the event horizon?

The last question I'm not really sure of, I have a feeling that I am way off.

Thanks