A Tangency Chain

Let $f(x) = e^x + x^2 - x$ and $g(x) = x^2 + ax + b$, with $a, b$ real.

1. When $a = 1$, find the intervals of monotonicity of $F(x) = f(x) - g(x)$. 2. Suppose the tangent line $l$ to $y = f(x)$ at $(0, 1)$ is tangent to $y = g(x)$ at $(1, c)$. Find $a, b, c$. 3. If $f(x) \geqslant g(x)$ for all $x$, find the maximum value of $a + b$.