How to formally model the “hesitation” in the hat-guessing puzzle and prove it by mathematical induction?
This isn't really an answer to your questions, but I think it would help in understanding the problem (or the way to answer your questions is) to write out a formal statement and proof of the solution. The proof you've presented doesn't say what is being proven, and as written is certainly incorrect. It states (final bullet point) that the inductive hypothesis is that $k$ people with $k-1$ black hats and $&gt; k$ white hats can tell that they all have white hats without any hesistation, but it proves that $k+1$ people with $(k, &gt;k+1)$ hats can tell with hesitation.

Writing out a more formal proof would also make it clear what modeling assumptions you require for it to work. For example, you would probably end up needing to prove "$k$ people with $(k-1,&gt;k)$ hats can all tell if they have white hats after $k$ moments of hesistation". If so, that means you would need some formal, common-knowledge notion of "moment of hesitation".

I hope that helps a little bit with question #1. (Also, the answers at math.stackexchange seem very good on this topic as well....) It might also help with #2, but I'm not sure what exactly you're asking -- I think the proof by induction would take care of this?