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I will show that the number of steps before a cycle is unbounded. Take any natural number $n$. Suppose that $p > n^{2^n}$, a prime. Then the chain starting at $p^n$ looks like: $p^n\to np^{n-1}\to f(n)(n-1)p^{n-2}\to f(f(n)(n-1))(n-2)p^{n-3}...$, so the chain hits a cycle after at least $n$ steps. This answers the second question. The first question seems more difficult.