Is there any way to remake the "Big Bang"?
Good question and something I've given some thought to. I haven't figured out a good way to represent the big bang in a realistic way (especially since Universe Sandbox doesn't simulate relativity).
This might be helpful in understanding why the Big Bang is difficult to simulate:
http://www.youtube.com/watch?v=nw5W3CszeAISomething that is fun to do (and helps to understand conceptually how all galaxies could be moving away from us) is to open up the system "Moons, Small Cluster" or even"Moons, Large Cluster" and then make gravity negative (so that everything repels each other).
You can make gravity negative by hitting the 6 key (although this key assignment may change in a future release) or by opening up the control panel and adding a negative in front of the value for gravity.
And is possible for dust to come together and make bodies?
No... Since dust is simulated as mass-less particles they don't attract each other. You can create many small bodies of a small size and with collision mode on "Combine Mode" they will come together to form larger bodies.
Try the Perpetual Motion activity under "Fun Thing to Do" to see this in action.