Prime factor inversion

Your function is OEIS sequence A008477, where I find the comment

For any n, the sequence n, a(n), a(a(n)), a(a(a(n))), ... is eventually periodic with period <= 2 [Farrokhi]. - N. J. A. Sloane, Apr 25 2009

The reference is to

M. Farrokhi, The Prime Exponentiation of an Integer: Problem 11315, Amer. Math. Monthly, 116 (2009), 470.

Q1: Yes, for example $p^q q^p$, where $p$ and $q$ are primes, is a fixed point (or products that consist of these). More generally you should be able to use permutations of order $2$, i.e products of disjoint transpositions.