Regarding the composition of permutations... So I have the permutations:
$$\pi=\left( \begin{array}{ccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
2 & 3 & 7 & 1 & 6 & 5 & 4 & 9 & 8
\end{array} \right)$$
$$\sigma=\left( \begin{array}{ccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
9 & 5 & 6 & 8 & 7 & 1 & 2 & 4 & 3
\end{array} \right)$$
I found $\pi\sigma$ and $\sigma\pi$ to be
$$\pi\sigma=\left( \begin{array}{ccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
5 & 6 & 2 & 9 & 1 & 7 & 8 & 3 & 4
\end{array} \right)$$
$$\sigma\pi=\left( \begin{array}{ccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
8 & 6 & 5 & 9 & 4 & 2 & 3 & 1 & 7
\end{array} \right)$$.
So my question is: did I do these correctly or did I mix them up (i.e, the permutation matrix I have for $\pi\sigma$ is actually $\sigma\pi$)?

You've done just fine - EXCEPT: the first is $\sigma \pi$ and the second is $\pi\sigma$.
$$\sigma \pi = \sigma\circ \pi = \sigma(\pi(i))\quad i = 1, 2, \ldots, 9\, $$ So we can only find $\sigma(\pi(i))$ after finding $\pi(i)$.
