If after a Big Crunch, the new singularity explodes in a Big Bang, would we get the same Universe all over again? Since black holes retain all the information they've stored, would we get an exact copy? And if not, why not? Where would the randomness come in? Is this the 9th time we're doing this . . . all this?
First off, let's start with the more common misunderstanding. The Big Bang was not an explosion of any kind. Popular science likes to depict it as an explosion because of the name "Big Bang" and also because it's more visually appealing than what the Big Bang actually was. The actual definition of the Big Bang is a little complicated, but suffice to say it refers to a moment in time (similar to the word "yesterday") and not an event in space.
Second, the Big Crunch scenario is considered an extremely unlikely scenario in modern cosmology. It is widely accepted that the expansion of the universe is accelerating and will continue to do so for the foreseeable future.
Even assuming black holes retained all information they captured, a black hole cannot explode. It is theorized that they can evaporate, but that is a far cry from any process violent enough to be call an explosion.
Continuing down the hypothetical, if all of the energy in the universe were "recompressed" into a similar state as at the beginning of the universe and a new Big Bang occurred, there is no reason for everything to come out looking the same. Matter and radiation would have to be recreated and the inherent randomness of quantum physics (not to mention even the minor fluctuations in the primordial background energy of that "new universe", which are expanded to relevant scales during inflation) allows for literally almost anything to happen. It could be entirely different.
This is likely not the 9th time we are doing this. This is either going to be the first time or an extremely large number-th time (as with any Big Bounce model)
It may be early to say it is not a cyclic universe. Everything we see around us (in spite of increasing entropy), appears to be cyclic. See atoms, solar systems, galaxies, clusters etc. Even the rate of expansion of universe has gone through cycles of acceleration and slow down. The last switch from slowed down expansion to accelerated expansion is expected to have taken place 5 billion years ago. To me, a cyclic universe makes more sense otherwise, why it would even happen once. Whether it will be an identical repeat or not is much more complex question. Presence of Inherent quantum randomness does not mean it is not regulated at some level. If it was not, then it would violate laws, and would never reconcile with them at a later point. There is something that knows that laws have been broken momentarily and enforces the return of order within short period of time. It is possible that no laws were broken in the first place, it may be just appearing (calculating) to us for a moment or so.