Pinched Between Combinations

Let $a \in \mathbb{R}$ and $f(x) = |x^2 - 1| - x^2 + ax$.

1. If $f(x)$ is an even function, find $a$. 2. Suppose the graph of $f(x)$ meets the line $y = 2x$ at exactly $2$ points in the first quadrant, with $x$-coordinates $x_1 < x_2$. Prove that