In the case of black holes, the radiation of stellar or galactic mass singularities is absolutely miniscule. Evaporation is only a "noticeable" process for very tiny black holes with the mass of an asteriod packed into the space of a proton.
As for what you can or cannot picture, that is your issue. I am just letting you know the basic phenomenon is much more broad and actually much more fundamental than a black hole event horizon membrane. The membrane and virtual pairs may begin (but not end) arguments and derivations or motivate theoretical preferences for resolving various issues, but it is misleading to call that imagined scenario the essence of the process. Physics teaching often suffers from "historical bias". Because some physicist first imagined things a particular way or convinced his peers a particular way, this is often the path used to motivate things to a popular audience. The truth is that after some thought and generalization it may be much less sensitive to the original motivating visual picture.
So, to answer (1), yes -- just an analogy. (2) would be correct if the answer to (1) were "no", but it isn't.
As I've referred to above, "capture" and "escape" of "virtual" particles is all a bunch of highly specific visualization related to a black hole or event horizon, but the actual result pertains to all accelerating reference frames and all spacetime curvature. Though Hawking himself might disagree with me, I find it pedagogically misleading to "explain" the possibility of this thermal radition in terms of processes only happening at a literal even horizon.
This is actually an interesting case of the strong principle of equivalence -- that gravity is locally indistinguishable from an accelerating frame of reference for all physical processes. (The weak principle of equiv is only about graviational forces, but the quantum vacuum is broader physics than that.) Specifically, you can derive Unruh radiation from quantum vacuum transformations *or* you get the same numerical temperature as starting from the idea that an accerelating reference frame "event horizon" is the same as a gravitational event horizon. I derived that latter in high school in the mid 80s, actually, to prove to myself that strong P of E held in this case. It's a relatively easy exercise in hyperbolic functions and basic calculus to compute the asymptotic trajectory of uniformly accelerating frame and back out the effective accerelation event horizon. Plug that in to Hawking's formula for a black hole and you get Unruh's result for acceleration. (They really call it Fulling-Davies-Unruh since it was done three times independently after the Hawking-Bekenstein results.)
I would agree with another responder here that not mentioning the thermal character of the radiation and words suggesting its monochromicity makes this particular result a little dubious, but I have not read the arXive article.
Now, its creation is a quantum state transition which has a "magical" quality in the same way that, say, a photon escaping an atom's electron shell does. There is no extended energy transport process at all. The electron makes a quantum jump simultaneously with the photon field of the world gaining a new photon traveling away. Indeed, with visible light, the wavelength of the photon -- hundreds of nanometers -- can easily exceed the spatial scale of the atoms electron shell, usually a few nm. So, the photon kind of just "appears".
As I mentioned above, one does not need a black hole for this -- all curved space should release thermal energy, though the rate is usually immeasurably small. Google Unruh effect and read about it in relation to the Sokolov-Ternov effect which has been observed since the 1970s. There is not perfect interpretational consensus about all this, though.
Work without a vision is slavery, Vision without work is a pipe dream, But vision with work is the hope of the world.