The Largest Integer Slope

Let $f(x) = \ln x$.

1. If $g(x) = f(x) - ax + \dfrac{1}{2}x^2$ has two extreme points, find the range of possible values of $a$. 2. If the equation $f(x) = m(x+1)$ with $m \in \mathbb{Z}$ has a real solution, find the largest possible integer $m$.