A Derivative Composition

Let $f(x) = \dfrac{2}{3}x^3 - mx^2 + m^2 x$ with $m \in \mathbb{R}$, and let $f'(x)$ be its derivative.

1. If $g(x) = f(x) - f'(x)$ has extreme values, find the range of possible values of $m$. 2. Let $h(x) = f'\left(\mathrm{e}^x\right) + f'(\ln x)$. If for every $m \in \mathbb{R}$ the inequality $h(x) \geqslant m^2 + k^2$ holds for all $x \in (0, +\infty)$, find the set of possible positive integer values of $k$.