To answer your questions first
This paper assume $f_i$ is sigmoid function $f_i(x) = \sigma(x) = \frac{1}{1 + e^{-x}}$.
Note that
$$ \frac{\partial \sigma(x)}{\partial x} = \sigma(x) (1 - \sigma(x)) $$
Since
$$ \begin{align} & f'{l_m}\big(\text{net}{l_m}(t - m)\big) w_{l_m l_{m - 1}} \ & = \sigma\big(\text{net}{l_m}(t - m)\big) \cdot \Big(1 - \sigma\big(\text{net}{l_m}(t - m)\big)\Big) \cdot w_{l_m l_{m - l}}, \tag{1} \end{align} $$
to find the maximum value of (1) with respect to $w_{l_m l_{m - 1}}$, we can calculate the derivative of (1) and find the point $w_{l_m l_{m - 1}}^$ where the derivative of (1) evaluated at $w_{l_m l_{m - 1}}^$ equals to 0, i.e. 804