Watson, so many questions you have! LOL
What is the speed of gravity? Instantaneous, the speed of light, or somewhere between?
Theoretically, the speed of light. otoh, Newtonian mechanics required that gravity be propagated instantaneously.
It's interesting to think about the orbit of the earth if the sun was magically removed from existence. If gravity moves at the speed of light, the earth would continue to orbit the missing sun for 8 minutes.
Yes. The same would be true if the Sun suddenly started moving away from the Earth; it would take 8 minutes for our planet to "feel" the change in gravity (i.e., for the gravitational radiation generated by the movement of the Sun's mass to reach the Earth).
Also, is gravity a particle or a wave (or both)?
Neither; gravity is a "field" effect,
somewhat analogous to a static electric charge. When an electric charge accelerates, it generates an electromagnetic radiation that propagates at the speed of light. Likewise, when a mass accelerates, it is theorized that a radiative effect is generated. The wave-particle duality of these equations is only a convenient conceptualization.
Light photons must have some miniscule amount of mass in order to be affected by gravity.
No. Photons (and the so-far only theoretical graviton) have no rest mass. The 'apparent' effect of a gravitational field on the 'line of travel' of a photon is due to the curvature of space, and does not require that a photon have any actual mass. A massless angel flying right by the Sun would also appear to deflect fom a straight line of travel (assuming, of course, that angels exist within our space-time continuum
).
If gravity itself is composed of particles travelling at least the speed of light, would it not also have some miniscule amount of mass?
No. "The graviton, as yet undetected, should be massless and stable, and have a spin of 2. Its interaction with matter would be extremely weak, and it is unlikely that present techniques are capable of verifying its existence."
Why aren't gravity particles affected by gravity?
As a matter of fact, in a way they are! "...we may say that any gravitational wave is itself a distribution of energy and momentum that contributes to the gravitational field of the wave."
(If they are, how does the gravitational force escape a black hole?)
Because the 'gravity-on-graviton' effect is not a real thing...it's a purely mathematical difficulty (called non-linearity) that makes solving Einstein's equations exactly so daunting.
One thing: so far, gravitational radiation has not yet been directly detected. One reason: in terms of order of magnitude, gravitational radiation is 10 trillion times less powerful than Newtonian (read 'ordinary') effects. For example, as Jupiter orbits the Sun it generates only 5.3 kW of gravitational radiation, far too little to be detectable using currently available techniques. Even the incredible pulsar in the Crab nebula does not generate enough grav-wave energy to be detected.
Now how this relates to this topic...I have absolutely no idea! LOL
Perhaps it simply demonstrates that we still have so much to learn, and that making unequivocal statements like "it must have happened that way, or be this way" betrays ignorance.
Craig