Part 1.) R. Stanley mentions on [page 59 (in the answer to exercise 24a, which is the identity mentioned in the OP)][1] in *Enumerative Combinatorics* 1 the following somewhat inconclusive fact:

> Some related results are due to Euler and recounted in $\S$303 of P.A. MacMahon, *Combinatory Analysis*, vol. 2, Cambridge University Press, 1916...

Searching the Euler archive showed a few similar results (but those were on the generating function of $\sigma(n)$ being related to the logarithmic derivative of the Euler generating function.

  [1]: http://books.google.com/books?id=EvJg1VjIGyMC&lpg=PA59