(1) $e^z$ about $z=πi$.
I simply wrote $e^z = 1 + z + z^2/2 + z^3/6 + \dots$ and noted that, since $e^z$ is entire, its series is valid for disc in infinite radius about $πi$.
(2) $\displaystyle\frac{z^2}{1-z}$ about $z=0$.
I wrote $$\frac{z^2}{1-z} = z^2(1 + z + z^2 + \dots).$$ This series converges when the modulus of $z$ is less than $1$. So we have an open disc with a unit radius and center at $0$.
Am I missing any key information?
Your first series is a power series about $0$, not about $\pi i$. You should have written\begin{align}e^z&=e^{z-\pi i+\pi i}\\&=-e^{z-\pi i}\\&=-1-(z-\pi i)-\frac1{2!}(z-\pi i)^2-\frac1{3!}(z-\pi i)^3-\cdots\end{align}
The other one is not a power series. You should have obtained$$z^2+z^3+z^4+z^5+\cdots$$