Here is an extended hint for the first conjectured formula.
As in Timothy Budd's answer let $w=w(q)$ denote the (formal) solution of $q=\frac{w}{f(w)}$.
Write the sum in the exponential above as
\begin{align*} \sum_{n>0} \frac{q^n}{n}\Big(\sum_{j>0}[z^{n+j}]\{f(z)^n\} w^j\Big) \end{align*}
and rewrite the inner sums as
\begin{align*} \sum_{j>0}[z^{n+j}]\{f(z)^n\} w^j =\frac{1}{w^n}\Big(f(w)^n -\sum_{j=0}^{n-1}[z^j]\{f(z)^n\}w^j\Big) -[z^n] \{f(z)^n\} \end{align*}
Note that \begin{align*} \sum_{n>0} \frac{q^n}{n}[z^n]\{f(z)^n\}=\log\big(f(w)\big) \end{align*} and that the remaining sum can be written as \begin{align*} \sum_{n>0} \frac{1}{n} [u^{n-1}]\Big\{\frac{1}{1-u} \big(1-\frac{f(uw)^n}{f(w)^n}\big)\Big\} \end{align*}
Summing up and evaluating the residue now gives the value $-\log\big(1-q\,f^\prime(w)\big)$
for this sum.
Finally, note that \begin{align*} \frac{1}{f(w)}\frac{1}{1-q\,f^\prime(w)}= \frac{q^2 w^\prime}{w^2}\;\;. \end{align*}