> We should see this as some inconsistency in how gravity scales with the mass of a black hole.
This inconsistency is a part of general relativity (even though gravitons themselves aren't). A black hole does not follow the GM/r^2 gravity law.
> If they are massive, and therefore subject to slowing, shouldn't gravity waves leaving a black hole be subject to some sort of doppler effect? Should we be looking for red/blueshifts in these waves?
Yes, there is a doppler effect.
> If gravitrons have mass and are subject to gravity, what brings that gravity? What sub-gravitron particle regulates gravity going in/to/out of the gravitron? This would require a new set of particles be created by non-gravitron massive objects (ie black holes) alongside the gravitrons. Like I said, too strange to exist.
Gravitons. Every now and then a pair of gravitons may exchange more gravitons (these are also virtual particles, so it's fine). This is why Feynman diagrams exist, so that you can not only calculate the interaction between two particles through a graviton, but also calculate the contribution of the lower-probability situations where more gravitons magically appear to transmit gravity between gravitons.
This is nothing new. Gluons (the carrier for the color/strong force) do basically the same thing. They are also bound by the force they carry, and thus gluons can interact with each other with more gluons.
Because these are virtual particles and only exist as a probability, this doesn't lead to infinite recursion. The secondary gravitons are improbable, and the tertiary gravitons more so, and so on, and the final series converges.