Consider the series $$ S_f = \sum_{x=1}^\infty \frac{f}{x^2+fx}. $$ Goldbach showed that, for integers $f \ge 1$, $$ S_f = 1 + \frac12 + \frac13 + \ldots + \frac1f $$ (this follows easily by writing $S_f$ as a telescoping series). Thus $S_f$ is rational for all natural numbers $f \ge 1$. Goldbach claimed that, for all nonintegral (rational) numbers $f$, the sum $S_f$ would be irrational.

Euler showed, by using the substitution $$ \frac1k = \int_0^1 x^{k-1} dx, $$ that $$ S_f = \int_0^1 \frac{1-x^f}{1-x} dx. $$ He evaluated this integral for $f = \frac12$ and found that $S_{1/2} = 2(1 - \ln 2)$ (this also follows easily from Goldbach's series for $S_f$). Thus Goldbach's claim holds for all $f \equiv \frac12 \bmod 1$ since $S_{f+1} = S_f + \frac1{f+1}$.

Here are my questions:

The irrationality of $\ln 2$ was established by Lambert, who proved that $e^r$ is irrational for all rational numbers $r \ne 0$. Are there any (simple) direct proofs?

Has Goldbach's claim about the irrationality of $S_f$ for nonintegral rational values of $f$ been settled in other cases?