How many of these will appear naturally in powerful AI systems?

Perhaps many! Maybe by trying to emulate a human brain we will end up recreating its flaws.

I am very excited in the progress of deep learning applied to symbolic, logical reasoning, like theorem proving. Theorem verification is easy and tractable, proving is not.

We can have heuristic algorithms come up with provably correct algorithms! That is vaguely analogous to a human writing a program then proving it correct. Now that will be useful.