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A couple of good responses so far. Let's expand the list of questions.

1. How do gravitons escape a black hole?

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A "graviton producing machine" inside the BH will not be able to send real gravitons out past the event horizon. (provided we remain in the limit where the BH solution can be thought of as a reasonable background)

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2. How does gravity escape a black hole?

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Gravity doesn't "escape" a black hole -- a black hole *is* a solution to the gravitational field equations.

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3. Is there anything special about the event horizon?

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Not locally -- you wouldn't notice anything special if you were in a box falling across the horizon. Outside the BH, though, it does have significance -- you'll never hear again from anything that crosses the horizon.

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4. If there is a graviton version of Hawking radiation, call it graviton-Hawking-radiation, then what is its strength?

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It should exist, and it should have a thermal spectrum just like any other field (provided again that the BH solution can be thought of as a good background -- i.e. the perturbative regime -- this should be an extremely good approximation for any reasonable BH, though).

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5. Is the strength of the graviton-Hawking-radiation basically the same strength as the gravity (gravitational acceleration) itself? (Set hbar, c, G to 1.)

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The temperature defining the thermal spectrum of gravitons should be proportional to the surface gravity of the BH (which will be inversely proportional to the mass of the BH).

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6. If so, or if not, what role does graviton-Hawking-radiation play in gravity?

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Due to the weakness of Hawking radiation, the gravitons can be thought of as tiny perturbations to the BH solution -- they shouldn't be significant in changing the causal structure of the BH solution.

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7. Since Hawking radiation comes from gravitional acceleration, does it not occur also inside the event horizon all the way down to the `singularity'?

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I seem to recall that a freely falling detector dropped into a BH isn't excited by Hawking radiation, but I could be mistaken.

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8. If not, then how does all the mass get from the `singularity' to the outside during the complete Hawking evaporation of a black hole via Hawking radiation?

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Hawking radiation is a property of the vacuum outside the BH -- there are some problems with thinking of Hawking radiation as something "escaping" from the BH. That said, there are some very big problems associated with "complete evaporation" of a BH that aren't properly understood by anyone.

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9. If so, how much of the mass of the black hole consists of Hawking radiation between the `singularity' and the event horizon?

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This is a sort of ill-defined question. The BH's mass is defined with respect to a "timelike Killing vector" that doesn't exist inside the BH. Energy and momentum inside the BH are only locally defined concepts.

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10. If you are accelerating `forwards' in space, does the Unruh radiation (similar concept to Hawking radiation, relating gravity to acceleration) hit you from the front or the back, or from all around, or what?

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It's isotropic.

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11. What are all the similarities, and differences, between gravitons and photons?

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This could lead to a huge, detailed discussion in and of itself. If you want to discuss it more, I wouldn't mind doing so, but I'll sidestep it for this particular post.

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12. (For genuine experts only) how much of the above is nonsense based on ignorance?

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They seem like reasonable questions, but as always a more detailed understanding reveals some "problems" to not be problems at all, and some questions not to be defined in the original sense that you imagined.