How can one of the particles on a pair of virtual particles created by vacuum fluctuations have negative energy? How does it lower black hole mass? Assuming Stephen Hawking’s 1975 “Particle Creation by Black Holes” :

“One might picture this negative energy flux in the following way. Just outside the event horizon there will be virtual pairs of particles, one with negative energy and one with positive energy. The negative particle is in a region which is classically forbidden but it can tunnel through the event horizon to the region inside the black hole where the Killing vector which represents time translations is spacelike.”
"Instead of thinking of negative energy particles tunneling through the horizon in the positive sense of time one could regard them as positive energy particles crossing the horizon on past-directed world-lines and then being scattered on to future-directed world-lines by the gravitational field."

 A: Note that energy (at least in this context, and in general too) is defined as the conserved charge associated with the generator of time translations. Outside the black hole, this generator is the timelike Killing vector. As you cross the event horizon from outside to inside, the Killing vector changes from timelike to lightlike to spacelike. So inside, the Killing vector is associated to 3-momenta, which can be positive or negative (unlike energy). See the nice review The Black Hole Information Problem by Polchinski.
I would also like to call attention to the perils of using the picture of virtual pairs too literally. Virtual particles are not real (hence the name) and their primary purpose is to act as mnemonics to jot down the amplitude associated to a particular Feynman diagram when working in perturbation theory. Hawking pointed this out in the same paper, in the very next sentence - "It should be emphasized that these pictures of the mechanism responsible for the thermal emission and area decrease are heuristic only
and should not be taken too literally."
A: I'm not sure at what level you are asking, and thus at what level you want an answer. This answer will be at the level of an informal guide for the non-expert.
A very useful way to calculate results in quantum field theory is the Feynman diagram method, which is a form of perturbation theory. Whenever people refer to virtual particles in the context of quantum field theory, this is very likely what they are doing. The term 'virtual particle' is then a reference to part of an integral describing the process in question. One speaks of 'virtual particle' for terms in which the propagator is a decaying exponential rather than an oscillating exponential. One may interpret this by saying that such a term is behaving mathematically a bit like a term for a propagating particle, but not quite like, so it is called a virtual particle. The energy and momentum of such terms (virtual particles) can have properties not allowed for real particles, such as transferring energy and momentum in the opposite direction to the (decaying) propagation. So in this sense they might be said to have negative energy.
As Hawking said in his paper, the statements in words about what the virtual particles are doing should not be taken too literally; they are simply a guide to some features of the mathematical equations. However, this picture can be cashed out quite well when one approaches the calculation a certain way. The particles in this case are mostly photons, by the way.
