You need to assume also that $f$ is non-isotrivial. (Think of $X=S\times F$).
On the other hand, you don't need the semi-stable assumption for any of these, except that the explicit bound you are asking about is better if you know semi-stability. Also, if there are only finitely many singular fibres, then the general fibre is smooth, right?
I also assume that you take $h$ to be the Hilbert polynomial of a fixed power of $\omega_{X_\eta}$.

For your first question in parentheses: The answer to "who showed this" depends on the context. One should probably mention the names Fujita, Kawamata, Viehweg, Kollár. Perhaps more.

For the second question the answer is "yes with $m=2$, as long as $f_*\omega_{X/S}^{\otimes n}\neq 0$". In fact, in that case $f_*\omega_{X/S}^{\otimes n}$ is ample. This was proved by Esnault and Viehweg in *Ample sheaves on moduli schemes*. Algebraic geometry and analytic geometry (Tokyo, 1990), 53–80,
ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991.

[**EDIT**]: I've just realized that you asked an explicit upper bound on $m$. The previous paragraph answers that. Here I "answered" a question I thought you were asking about an upper bound for the degree:

There exists an upper bound on the degree, which slightly depends on your assumptions, but they all come down to be something like $(const)\cdot \mathrm{rank}(f_*\omega_{X/S}^{\otimes n})\cdot (2g(S)-2+\delta)$, where $\delta$ is the number of singular fibres. Results of this kind have been obtained by
Bedulev-Viehweg, Viehweg-Zuo$^1$ (J. Algebraic Geom. 10 (2001), no. 4, 781–799),
Kovács.

In fact, there have been a lot of results related to these questions and various generalizations. You may start with the survey Kovács, Sándor J.
*Subvarieties of moduli stacks of canonically polarized varieties: generalizations of Shafarevich's conjecture*, Algebraic geometry—Seattle 2005. Part 2, 685–709,
Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009, and look at the references.

These results have strong implications to hyperbolicity of moduli spaces and generalizations of the Shafarevich conjecture to higher dimensional varieties. See for instance a paper by Kovács-Lieblich.