Part 1.) R. Stanley mentions on page 59 (in the answer to exercise 24a, which is the identity mentioned in the OP) in Enumerative Combinatorics vol. 1 (exercise is 78a in later edition) 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.
