For starters, you're wrong in phrasing your question on a couple of levels... such as "...one section of the cosmos...", "formed all the matter in the universe"... The Big Bang is[i the cosmos and [i]is] the universe. Nothing (which is impossible for our finite minds to grasp) existed before the event. Time began at the moment of the event... time did not exist prior.
Cosmologists can't explain exactly how long ago the event took place... they can describe (via equations) history back to 10^-43 seconds or so... (Planck Time) but not further... prior to that all four fundamental forces were unified (as well as additional forces, some propose).
The Big Bang is certainly a misleading misnomer (sorry for the mixed metaphors). There was no "Bang" as such... it was more accurately an "expansion" but an expansion that exceededthe speed of light for a very short time.
By the way... the type of expansion characteristics inherent in most models indicate the universe not to be a sphere at all, but rather is flat in its geometry.