a) entanglement does not transfer information faster than light. Why? if i send entangled pairs of photons from a to b and c, and b and c detect these photons, the photons took time to reach b and c. If b does something to the photon, the entanglement is lost. If b and c measure they will know the state the other one received, however they can not influence what is received in the other place, so sorry guys, no FTL transmission of information
b) What is weird about entanglement is actually not so much it statistical property of the correlation. If a packs a white and black marble in two packages and mixes the packages and sends them out, the result from the viewpoint of b and c will be the same - each one will know which marble the other one received. The weird property is that the state is prepared in a way that the two possibilities are quantum states, which can be subject to phase shift, transitions etc, and are "collective" in that sense that b and c can transfer their state to particles (and possibly create further entanglement) - the basis for Quantum key distribution networks - and that the information which exists exists only in the form of a shared posteriori observation. i.e. the classical marble can be looked at without destroying the correlation, while a quantum entangled photon will be entangled with your measurement apparatus when looking at it.
c) what these guys did-AFAIU (my topic was very far away) is to create a model system of a black whole, which tries to represent a black whole in a way which we assume it is, observed some properties which can be predicted from this model (temperature of emitted radiation), and checked for some others - correlation, where they found correlation which they interpret as entanglment.
d) While did not look into the details, i can say from my own experience that such experiments are tricky, and i find the interpretation a little vague. But i have to look closer. I did use quantum state/operator tomography, which usually is the benchmark measurement when you want to prove entanglement, or properties of the superoperators describing your quantum operations. I understand that this may not be possible in this case, which is why one can go for other phenomenological approaches
e) One should be careful. Proving entanglement is not so simple (Look for entanglement measures), and proving that is actually *survives* the event horizon, instead of being created there, may be very nontrivial. It could very well be that non-entangled state are transformed in entangled states to some degree.