Solution:
(1) Since the equation of line $l$ is $$\rho\cos\left(\theta - \frac{\pi}{4}\right) = -4\sqrt{2},$$
it follows that $$\rho\cos\theta\cos\frac{\pi}{4} + \rho\sin\theta\sin\frac{\pi}{4} = -4\sqrt{2},$$
thus, the Cartesian coordinate equation of line $l$ is $$\frac{\sqrt{2}}{2}x + \frac{\sqrt{2}}{2}y = -4\sqrt{2},$$ which simplifies to $x + y + 8 = 0$.
(2) Since the parametric equation of curve $C$ is $$\begin{cases} x=8\tan^{2}\theta \\ y=8\tan\theta \end{cases}$$ (where $\theta$ is the parameter, $\theta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$),
and $P$ is a point on curve $C$ and $Q$ is a point on line $l$,
the distance $d$ from point $P(8\tan^2\theta, 8\tan\theta)$ to line $l$ is:
$$d = \frac{|8\tan^{2}\theta + 8\tan\theta + 8|}{\sqrt{2}} = 4\sqrt{2}\left|\left(\tan\theta + \frac{1}{2}\right)^2 + \frac{3}{4}\right| = 4\sqrt{2}\left(\tan\theta + \frac{1}{2}\right)^2 + 3\sqrt{2},$$
thus, when $\tan\theta = -\frac{1}{2}$, $|PQ|$ reaches its minimum value of $3\sqrt{2}$.
Therefore, the minimum value of $|PQ|$ is $\boxed{3\sqrt{2}}$.