Does $[\mathfrak{h},[\mathfrak{h},\mathfrak{h}]]=0$ imply $[\mathfrak{h},\mathfrak{h}]=0$?
No, it is not true in general. Let $\mathfrak{g}$ be the $3$-dimensional Heisenberg Lie algebra with basis $(x,y,z)$ and Lie brackets determined by $[x,y]=z$. Then $\mathfrak{h}=\mathfrak{g}$ satisfies $[X,[Y,Z]]=0$ for all $X,Y,Z$, but the Heisenberg Lie algebra is not abelian.