If any configuration of matter can fall into a black hole and hit the singularity, and ditto for the big crunch, and there is time reversal CPT invariance, does it mean anything can pop out of the big bang singularity, and there is unpredictability at the big bang?
How do we explain the very low entropy of the big bang then?
The Big Bang, in the usual meaning of the term, is the zero time limit of the FLRW metric. If you make a few simplifying assumptions about the universe, principally that it's isotropic and homogenous, you can solve the Einstein equation for the universe as a whole, and the solution is the FLRW metric. This describes the evolution of the universe with time.
If you take this metric and wind the time back towards zero you find the temperature and density get bigger and bigger until at time zero the temperature and density become infinite. Amusingly, the size of the universe at this point is infinity times zero, and whenever we encounter expresions like this in Physics it means we've pushed our mathematical model beyond the limits of its applicability and it's no longer predictive.
So yes, if you stick to the FLRW metric there is unpredictability at the Big Bang.
However most of us believe that quantum mechanics will take over at times very close to zero and this will stop the density and temperature becoming infinite. There is no universally accepted theory of quantum gravity yet, let along a quantum description of the Big Bang, so we don't know what will happen, but we expect it to be predictable. If you believe Loop Quantum Cosmology this predicts the universe will shrink to a minimum size then bounce back again, but this is speculative.
Anyhow the Big Bang isn't a time reversed black hole. Have a look at John Baez's article for more details.