For a homogeneous Markov chain on a finite state space $X$ with probability transition matrix $P$ and initial distribution $\mu$, what is the value of
$$P(\omega_n = x^{1} \mid \omega_0 = x^2, \omega_{2n} = x^{3}), $$
where $x^1, x^2, x^3 \in X$?
I am aware of the Markov property, so I think I can rewrite it as
$$P(\omega_n = x^{1} \mid \omega_0 = x^{2}) $$
because the future states don't matter. Then I guess the answer is just $\mu p^{(n)}_{x^2, x^1}$? Is this correct?
Incorrect. If you know you end up at $x_3$ it absolutely can affect the prob distribution of where you were in the past. The Markov property said, conditioned on a past state, then further-distant past does not matter. It never said that the future does not matter.
In your example, suppose $x_2$ can reach $x_1$ and $x_3$, but $x_1$ cannot reach $x_3$. Then conditioning on a future at $x_3$, the prob of being at $x_1$ is $0$ (since you couldn't have reached $x_3$ from $x_1$).
To actually solve your problem, here's a hint: use the definition of conditional prob $P(A \mid B) = P(A \cap B) / P(B)$. In fact, use it multiple times.
Can you finish from here?