When you ask "Why is gravity such a unique force?" then you should know that in the framework of General Relativity gravity is not a force at all. In General Relativity energy (for example the mass of an object) cause curvature. The movement of other objects is influenced by this curvature - they travel along the path of shortest distance between two points (called geodesic), which is not necessarily a straight line. Thus, objects are "free falling" as long as "real forces" like electromagnetism prevent them from going on a geodesic. For example, the electromagnetic force is responsible for the repulsion between the ground on which you stand and your shoes. This force prevents you from going on the geodesic (falling towards the centre of the earth). We interpret this process wrongly as a gravitational force pulling us down.
That's the picture of General Relativity. The electromagnetic, the weak and the strong force are the real forces. The formulation of these forces as Quantum Field Theories is called the Standard Model. This is the model you pictured.
But Quantum Field Theory is only valid in a flat spacetime, without curvature. Also, it does not include gravity at all and it does not postulate gravitons. Thus, the Standard Model is only an approximation. 343
We know that the spacetime around the Earth is curved. Working with the Standard Model we approximate this curved spacetime by flat spacetime. This approximation works out pretty well since the curvature caused by the Earth is not too great. So, we neglect gravity altogether in the Standard Model. It only works, because the effects of gravity are so small when conducting experiments on the Earth.
Both General Relativity and the Standard Model are tested experimentally and they give very good results. But both theories are incomplete: the Standard Model does not include gravity at all and General Relativity is not a Quantum Theory. This is weird since our understanding of the world suggests that every fundamental theory should be a Quantum Theory.
Unfortunately, nobody figured out how to merge these two theories into one consistent theory. Every attempt to do so is pure speculation at this point without any experimental support. One such speculation is to introduce the graviton, a supposed mediator of the gravitational force in the framework of Quantum Field Theory. But, in General Relativity gravity is not a force at all and it is not clear how one should bring together this two notions. As far as I know, attempts to do so require radical rethinking of our basic understanding of physics - for example String Theory. Maybe an expert on String Theory can elaborate on that. 651