Did big bang create gravity? What role gravity is assumed to have played in the formation (starting from the big bang) of large structures of our universe and what other important physical mechanisms and processes probably led to the structure we observe today?
Gravity has played a dominant role in the development of the large-scale structure of our universe.
The largest structures of matter in our universe (most of it dark matter) grew out of over-densities in the primordial matter distribution that emerged shortly after the big bang. Dense areas began collapsing under their own gravity, and over time as their mass grew they could pull in matter from larger and larger volumes that had initially been moving away in the universe's expansion.
A powerful analytic treatment of this gravity-induced growth of structure is called cosmological perturbation theory. At small spatial scales and later times this treatment can become tedious or break down entirely, and numerical N-body simulations such as the Millennium Run can provide additional information about the growth of structure. Essentially, the only physical process involved in these simulations is gravity (although the initial conditions require insight from other physical processes).
At the scales of clusters of galaxies and within single galaxies, other processes become important in addition to gravity. Ordinary (non-dark) matter can be heated and emit light to cool itself, and this must be accounted for. Gas also forms stars, which heat surrounding gas and may explode as supernovae. Large black holes that grow at the centers of massive galaxies heat gas to large temperatures in their accretion disks, and the light emitted from these disks can potentially influence the circum-galactic and even circum-cluster gas. All of these physical processes in non-dark matter can also feed back gravitationally into the evolution of the dark matter. Accounting for all of these effects in cosmological simulations is a topic of ongoing research.
As to whether the "Big Bang created gravity," I think physicists might disagree about whether this is true, false, or a matter of semantics or philosophy that lies outside of physics. Some may take a view that we as scientists are not qualified to speak about matters of the creation of physical laws at the time of the Big Bang, but only about observable and testable-in-principle matters that came into existence after the Bang. Some others would say that such questions are approachable from the point of view of the Multiverse. Some others might say that gravity, as an element of General Relativity, "just is" and it is entirely improper to speak of its creation. There are most likely other scientific points of view.