Depth reduction for Boolean circuits This result by Tavenas, Koiran and others show that any polynomial computed by a circuit of size $s$ is computed by a depth-4 homogenous circuit of size $s^{\sqrt{d}}$.

Are there any similar results for Boolean circuits or do we know why such a thing is not possible?
Unfortunately, the best known depth reductions for Boolean circuits only work for very restricted classes of circuits. Valiant (Valiant; Viola) proved that a circuit of size $O(n)$ and depth $O(\log{n})$ can be computed by a depth-3 circuit of size $2^{O(n/\log\log{n})}$. Also, Valiant showed a similar depth reduction for linear size series-parallel circuits (a natural subclass of circuits), see Calabro.

Note that the number of gates in a depth-3 circuit computing a random function is $\Theta(2^{n/2})$ (Dančík; Sergeev), while the best lower bound we can prove is only $2^{\Omega(\sqrt{n})}$ (Håstad; Håstad, Jukna, Pudlák; Paturi, Pudlák, Zane; Boppana; Paturi, Pudlák, Saks, Zane; Meir, Wigderson).