How does gravity force get transmitted? How does gravity force get transmitted? 
It is not transmitted by particles I guess. Because if it was, then its propagation speed would be limited by the speed of light. If it is not transmitted by particles how is it transmitted then? 
 A: 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.
A: 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
A: 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.
A: In some respects, static gravity is similar to static electricity in the sense that both follow the inverse square law. The major difference between the two is that similar charges attract in gravity, but repel in electricity. Also as far as we know, there is only positive mass (charge) in gravity. If negative mass existed, it would in fact be repelled by (normal) positive mass according to the inverse square law anyway, and will not be found around. The similar charges attract issue can in fact be easily overcome- by putting im for the mass m, where i is the imaginary number unit. 
In the dynamic case where we have motion, we have another similarity between the two forces- the effects of the forces propagate at the fixed speed of light. I think this is a big hint that the two are closely related. If we then take such delay into consideration, then look to modify the forces accordingly, we can use the retarded potential integral. If we do this for electric charges, we can recover the complete set of Maxwell equations, from the static Coulomb force alone. As we know, these equations are relativistic. 
The same can be done to gravity and Newton's law to obtain the so called gravito-magnetic equations. These are similar to Maxwell equations in form, and shown to be derivable from GR in the weak limit or linear gravity. These equations work not in 4D space-time, but like Maxwell in 3+1 dimensions(flat space and time). In this case, it is possible in principle to introduce gravitons very much like the photons.
The question now is that GR is non-linear and the gravito-magnetic equations, like Maxwell equations, are linear. This could stop the flow of this argument from going any further. In my opinion however, the reason GR is nonlinear is the same as Maxwell equations becoming non-linear, or the Schrodinger equation becoming nonlinear, in material media. It's a result of double counting.. we take the energy in a field and mix it with that in a mass- resulting in a multiplicity of counting and a nonlinearity in the variables of the resulting equations.
Having said all that, it must also be recalled that the photon is a quantity of energy representing the way electrically charged bodies exchange energy, and as such, is closely related to the energy levels in an atom. Thus, for the graviton to have a use and a meaning, similar energy exchanges and levels are needed in gravitating systems. Gravitating systems mainly exchange mass-energy much more than any other form of energy.
