Can neural network process randomness?
Purely randomness is not possible but in real-world the most randomness that a neural network can process is enough to discover an approximation pattern; the precision depends on the model of the context (exactitude of the Digital Twins) and processing power.

To extract the exact pattern from the pure randomness (I think) is impossible even with a quantum computer and even having all the time and resources of the universe. The reason is that nothing in the universe happens randomly, every effect has a cause. So regarding to the third case (theoretically) if you can reproduce every cause of a "twin universe" you can predict the future.