Let $(F(x)\frac{d}{dx})^n=\sum_{i=1}^n H_{n,i}(F, F', F^{(2)}, \ldots , F^{(n)})\frac{d^i}{dx^i}$. I'm curious about the exact formula for $H_{n,i}(y_0, y_1, \ldots , y_n)$. What is known about it?

Let me start by giving the formula $$H_{n,l}(y_0,y_1,\dots,y_n)=\sum_{(k_1,k_2,\dots,k_{n1})\in P_{n,l}}\frac{y_{0}}{l!}\prod_{j=1}^n (j+1k_1\cdotsk_j)\frac{y_{k_j}}{k_j!}$$ where, $P_{n.l}$ is the set of $(k_1,k_2,\dots,k_{n1})\in \mathbb Z_{\geq 0}^{n1}$ which satisfy $k_1+\cdots+k_{n1}=nl$ and $k_1+\cdots+k_i\le i$ for all $1\le i \le n$. In this form this is due to L. Comtet:
This has an OEIS entry, where you can find more references. In particular see Bergeron, F. and Reutenauer, C., Une interpretation combinatoire des puissances d'un operateur differentiel lineaire, Ann. Sci. Math. Quebec. 11, 269278 (1987) for a combinatorial interpretation in terms of forests of rooted trees. The analogous expansion for the multivariable case is treated in "Universal expansion of the powers of a derivation" by M. Ginocchio. There are also generalizations to the noncommutative case that are described in his other article "On the Hopf algebra of functional graphs and differential algebras". 

