Miscellaneous quizzes.

You do realize that in that case, the 2nd person can't deduct what he did, right?
You got it close though, just need to cover a few more possibilities.

Let me rephase my answer:

 Spoiler

let's say H=hat and N=no hat
In the case of NNH, the 3rd person will see immediately that the first 2 person doesn't have hat and say "I have a hat" immediately".
In the case of HHH, HHN, HNH, HNN, NHN, NHH : the last person sees the one of the first 2 have a hat , and doesn't talk.
about 10-30 seconds later, the 2nd person would know that the 3rd person sees at least 1 hat in either him or the 1st person which is why the 3rd person didn't have the hat.
So if the hat were NHH, NHN, the 2nd person would know that he have the hat.
and he wouldn't talk when it's HHH, HHN, HNH, HNN.
about a long while later (almost time's up), neither the 2nd or the 3rd person talked, the first person would know, it's because he have the hat, that's why the previous 2 are afraid to say "I have a hat", which covers the case HHH, HHN, HNN, HNH

for 2nd question from Darkmaniak:

 Spoiler

gold chest