Could someone define "universe"? Do you mean *our* universe the Milky Way, and all objects within it (synonymous with our "galaxy"), or do you mean our universe plus all of space outside that universe and all objects within it as we know it thanks to modern technology, and also all object we don't know about but can probably agree exist due to space being infinite?
Mind you I'm no physics expert, but if by "Universe" we mean "Milky Way" then yes, there may have been a Bang that began this particular galaxy and the time within it, and maybe also a smaller Bang that created the solar system we now live in, just the same way we can now observe this happening in other galaxies. But for one Bang to have created all of space and time? Is that even possible?
Also, are other objects really expanding away from us, or are we pulling away from them?