Three Tangents, Exactly

Let $f(x) = x^3 - 3x^2 - x + 3$.

1. Compute $\displaystyle\sum_{i=1}^{7} f\!\left(\frac{i}{4}\right)$. 2. If the curve $y = f(x) + a\ln x$ (for $x \in (3, 4)$) has two mutually perpendicular tangent lines, find the range of possible values of $a$. 3. Let $M$ be a point to the right of the $y$-axis. If exactly $3$ tangent lines to the curve $y = f(x)$ pass through $M$ (and this characterizes $M$), find the set of all such points $M$.