I'm very confused about how to do part (a) of the question. I've set up a systems of equations for the stationary distribution based on the transition matrix but thats too messy. I know there's a more elegant way of doing it. I think the proof of reversibility has something to do with irreducibility of the MC. But I'm not sure how to wrangle it out. After the proof, I think there will be some kind of substitution based on the conditional definition of a SD and the detailed balance equations.
Let $\pi_{ijk} = \alpha p_i^2 p_j$, where $\alpha$ is some normalization constant that ensures that the probabilities add up to one. To establish reversibility, we must show that for all states $ijk$ and $abc$: $$ \pi_{ijk} P_{ijk,abc} = \pi_{abc} P_{abc,ijk} $$
To this end, notice that there are three types of nonzero transition probabilities: \begin{align*} P_{ijk, ijk} &= p_i \\ P_{ijk, jik} &= p_j \\ P_{ijk, ikj} &= p_k \end{align*}
It suffices to verify the reversibility equation for the last two types of probabilities. Indeed, observe that: \begin{align*} \pi_{ijk} P_{ijk,jik} &= (\alpha p_i^2 p_j)(p_j) = \alpha p_i^2p_j^2 = (\alpha p_j^2 p_i)(p_i) = \pi_{jik}P_{jik,ijk} \\ \pi_{ijk} P_{ijk,ikj} &= (\alpha p_i^2 p_j)(p_k) = \alpha p_i^2p_j p_k = (\alpha p_i^2 p_k)(p_j) = \pi_{ikj}P_{ikj,ijk} \\ \end{align*} as desired. $~~\blacksquare$