A Derivative Meets a Parameter

Let $\mathrm{e}$ be the base of the natural logarithm, let $f(x) = \mathrm{e}^x + \sin x - 2x$, and let $g(x)$ be the derivative of $f(x)$.

1. Find the equation of the tangent line to the curve $y = f(x)$ at the point $(0, 1)$. 2. If for every $x \in \left[-\dfrac{\pi}{3}, 0\right]$ we have $x \cdot g(x) \geqslant x^2 + m$, find the range of possible values of the real number $m$.