If one believes in the theory of gravitons then by viewing a black hole you see gravitons affect photons. This in turn leads to the conclusion that force carrier's mass equivalences allow them to be affected by gravitons. However if all force carriers were affected by gravitons then shouldn't gravitons be affected by gravitons? And if this were the case why would black holes exist? (Since the gravitons would be sucked into the black hole). This relies on the assumption that mass equivalence is a part of all force carrier particles, not just photons. Side note is that gravitons are theoretical.
There is really too much going on in this question - basically it is starting at point A and trying to get to point E without even stopping to take a breath at points B, C, and D, each of which you could spend years studying. Nonetheless, perhaps I can point in the right direction...
First, force carriers like photons are affected by gravity, quantized or not. We know this due to gravitational lensing, for instance.
Second: Yes, gravity affects itself. It is precisely this feedback that makes general relativity so much more difficult and nonlinear compared to electromagnetism. The presence of gravitating matter adds to the local stress-energy, which in turn increases the amount of gravity. This is why the gravitational mass (measured from the orbits of distant objects) of a neutron star is different from the sum of the rest masses of its constituents.
Third, regarding how we can feel gravity from a black hole, the problem is you seem to think of gravitons emerging from the singularity, passing out of the event horizon, and interacting with external matter. However, when dealing with virtual particles, one needs to be more cautious about assigning them definite trajectories as though they were normal, classical particles. The pull of a black hole is due to the overall curvature of spacetime (or the state of excitation of the graviton field, if you really want to push in that direction) - thinking of it as being sourced by a point can be misleading.
This is completely hand waving as gravitation is not in my expertise. Nevertheless:
It is true that in the classical black hole photons are trapped within the event horizon.
1) If we suppose that the graviton is similar to the photon and can be described by Feynman diagrams then the difference in attraction between photons and black holes and gravitons and black holes in the strongest diagrams ( first order) will depend on the product of the two couplings , a_electromagnetic*a_gravitational.
We get an order of magnitude 10^-3*10^-45
gravitation with gravitation, which would be the case to trap the gravitons will be of order
I do not know the actual numbers but seems to me the difference is large enough that a black hole would eat gravitons at lifetimes much larger than the existence of the universe.
2)One has to consider quantization. If a black hole were charged, virtual photons would leave and interact with matter outside the horizon of the black hole. It is the same with the gravitational "charge" of the black hole: virtual gravitons beyond the horizon will interact with matter beyond the horizon and thus we see the gravitational field of a black hole.