A Double Quantifier Inequality

Let $f(x) = |x - m|$ and $g(x) = x|x - m| + m^2 - 7m$.

1. If the equation $f(x) = |m|$ has two distinct real roots in $[-4, +\infty)$, find the range of possible values of $m$. 2. If for every $x_1 \in (-\infty, 4]$ there exists $x_2 \in [3, +\infty)$ with $f(x_1) > g(x_2)$, find the range of possible values of $m$.