We introduce the $(m,g,a_{0})$-Bell polynomials to construct the graded pointed Hopf algebras $\mathcal{H}_{\operatorname{FbB}}^{(m,g)}$ and their Hopf Ore extensions, which contain the well-known Fa\`a di Bruno Hopf algebra. We then give the isomorphism theorem of those Hopf algebras and study their Hopf subalgebras.  The noncommutative versions of $\mathcal{H}_{\operatorname{FbB}}^{(1,g_{1}^{r})}$ are determined. Finally, we use some special Lyndon words to construct several free bialgebras containing some basic combinatorial bialgebras and the noncommutative Fa\`a di Bruno bialgebra as quotients.

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