I did think along these lines initially. See this question: mathoverflow.net/questions/89304/…. I don't actually care about "for all sufficiently large n" statements, so I'm happy to take n even, say, in which case you have a large transitive subgroups such as $S_\frac{n}{2} \wr C_2$.

Yes I think so.

Why do you have that ergodic belief?

Reading up, mainly, and learning about Fourier analysis and representation theory.

It follows from en.wikipedia.org/wiki/Law_of_total_expectation. Suppose there are $m$ men remaining, and that they all go to fetch their umbrellas. This adds 1 to the rounds-count, hence the initial 1. After the round, with probability $p_k$, there are exactly $k$ men remaining, and the expected number of rounds to get rid of them all is $t_k$.