sequences and series - Strategies for evaluating sums $sum_{n=1}^infty frac{H_n^{(m)}z^n}{n}$

I'm looking for strategies for evaluating the following sums for given $z$ and $m$:
$$\mathcal{S}_m(z):=\sum_{n=1}^\infty \frac{H_n^{(m)}z^n}{n},$$
where $H_n^{(m)}$ is the generalized harmonic number, and $|z|<1$, $m \in \mathbb{R}$.

Using the generating function of the generalized harmonic numbers, an equivalent problem is to evaluate the following integral:
$$\mathcal{S}_m(z) = \int_0^z \frac{\operatorname{Li}_m(t)}{t(1-t)}\,dt,$$
where $\operatorname{Li}_m(t)$ is the polylogarithm function, and $|z|<1$, $m \in \mathbb{R}$.

Question 1: Are there any way to reduce the sum to Euler sum values, given by Flajolet–Salvy paper?

Question 2: Are there any way to reduce the integral to integrals given by Freitas paper?

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The case $m=1$ and $z=1/2$ was the problem 1240 in Mathematics Magazine, Vol. 60, No. 2, pp. 118–119. (Apr., 1987) by Coffman, S. W.
$$\mathcal{S}_1\left(\tfrac12\right)=\sum_{n=1}^\infty \frac{H_n}{n2^n} = \frac{\pi^2}{12}.$$
There are several solutions in the linked paper.

The more interesting case $m=2$ and $z=1/2$ is listed at Harmonic Number, MathWorld, eq. $(41)$:
$$\mathcal{S}_2\left(\tfrac12\right)=\sum_{n=1}^\infty \frac{H_n^{(2)}}{n2^n} = \frac{5}{8}\zeta(3).$$

We know less about the evaluation. At the MathWorld it is marked as "B. Cloitre (pers. comm., Oct. 4, 2004)". This value is also listed at pi314.net, eq. $(701)$. Unfortunately, I don't know about any paper/book reference for this value. It would be nice to see some.

Question 3: How could we evaluate the case $m=2$, $z=1/2$?

It would be nice to see a solution for the sum form, but also solutions for the integral form are welcome.