Let $l,\ m$ be the tangent lines passing through the point $A(a,\ a-1)$ on the line $y=x-1$ and touch the parabola $y=x^2$.
Note that the slope of $l$ is greater than that of $m$.

(1) Exress the slope of $l$ in terms of $a$.

(2) Denote $P,\ Q$ be the points of tangency of the lines $l,\ m$ and the parabola $y=x^2$.
Find the minimum area of the part bounded by the line segment $PQ$ and the parabola $y=x^2$.

(3) Find the minimum distance between the parabola $y=x^2$ and the line $y=x-1$.