I'm no expert in Astrophysics, Quantum Mechanics, or any other big-fancy title science fields, however it seems that you are asking questions that there is no finite answer too with our current knowledge of the Universe, in fact, I'm pretty sure we don't even have much information about our own galaxy.
However it does spark interesting conversation, and the what-if theories and scenarios intrigue me so I'll attempt to answer you last question if I understand it correctly.
If I understand correctly, you're asking what would be the resulting universe if a Big-Bang Explosion where to fire then collapse, fire then collapse, rinse and repeat, right?
I'm not entirely sure that's possible, an explosion on the scale of the big-bang is almost unimaginably powerful. Sure it can be roughly estimated through mathematics and so-forth, but nobody has ever experienced such a large scale display of force, and if one did occur I doubt they, or anyone would live to talk about it.
My point is... the big bang probably couldn't generate enough reverse pull to collapse back onto itself due to the force of the explosion in the first place.
I could, however, be entirely mistaken.
Now, let's say it where possible...
Then the first explosion would no doubt fire matter/radiation from the core mass, causing a 'ring.' After this burst It would also take time to re-collapse into a singularity, thus giving a small amount of time for the first release of matter to travel some distance. Now... the second explosion would occur, creating a gap space void of material. If this where to continue the universe would look somewhat like the age lines of a tree.
If this DID happen, we could possibly estimate the true age of the universe by counting the lines. (A bit of humor there.
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