so the singularity at z = 0 is a simple pole? is that correct to say?
I'm uncertain as to why it is zcot(z) and not just cot(z)/z^4?
No. The previous poster is giving you a hint as to what the smallest integer value of $\displaystyle n$ is such that $\displaystyle \lim_{z \to 0} \left( z^n \frac{\cot z}{z^4}\right)$ exists and is finite.
Also, your problem said to "find and classify all the singularities". Since $\displaystyle cot(z)= \frac{cos(z)}{sin(z)}$ this function has a singularity wherever sin(z) is 0.