How does gravity force get transmitted?
When you learn about gravity at college you're almost always considering situations that are static i.e. they don't change with time. For example you'll learn early on that the gravitational potential of the Earth is given by Newton's equation and this doesn't have any dependance on time. Later on you'll learn that General Relativity gives a more accurate description of the gravity of the Earth (or more usually of a star) called the Schwarzchild metric. This does have a time variable in it, but the metric itself it is independant of time just like Newton's equation.
To address your question about the propogation of gravity you have to ask questions like "what would happen to satellites if the Earth suddenly disappeared?". When the Earth is present the satellites are happily orbiting because of Earth's gravity. If the Earth suddenly disappeared would the satellites instantly veer off in a straight line, or would it take some time before they reacted to the Earth's disappearance.
The answer is that they would take some time to react because the change in the Earth's gravitational field would propagate at the speed of light. The change propagates by gravitational waves and these travel at the speed of light.
A gravitational wave is basically a disturbance in the curvature of space. Consider this analogy. A water wave is a disturbance in the surface of water. Suppose you had a model of the Earth floating on a pond and you suddenly pulled it out to leave a hemispherical dimple. Waves would flow into the dimple then spread out across the pond. A duck floating some distance away wouldn't know immediately that the model Earth was gone: it would only know when the waves reached it.
This is a somewhat dodgy analogy (I can hear the general relativists screaming already!) so don't take it too seriously. Apart from anything else gravity waves are mathematically very different from water waves. Still, I hope it gives the general idea. Changes in gravitational fields propagate by gravitational waves, and these move at the speed of light.
You mention particles. The description about is a classical one, and you might ask how quantum mechanics views the situation. After all, radio waves are a classical description and quantum mechanics views them as made up from particles called photons. Well you can describe a quantum gravitational field as being made up of particles called gravitons. However it is not at all clear that gravitons are a good description of quantum gravity. No-one has ever observed them, but then it would take energies far far higher than those attainable at the LHC to see gravitons, so it's no surprise they haven't been observed yet. If gravitons do exist they will travel at the speed of light just like photons.
Propagation of gravitational force is limited by the speed of light. In fact transmission of any kind of information is restricted by the speed of light. According to the general theory of relativity gravitational force propagates at the speed of light
It depends on how you think about gravity. In the framework of general relativity (the most complete, accepted paradigm), then gravity isn't a 'force' in the classical sense---but is instead the results of the geometry of space-time. Energy/mass curve spacetime; other bodies react to that curvature in their motion. Thus there is no force-carrier.
If you consider gravity in a particle-physics framework (which we don't have a complete model for, but many people are working on models of such a 'quantum gravity'), then gravity is believed to be conveyed by the spin-2 graviton.
In both cases changes in gravity propagate at the speed of light.