Let $G$ be an abstract group. The Malcev completion $\widehat{G}$ of $G$ (over $\mathbb{Q}$) is the set of group-like elements in the complete Hopf algebra $\widehat{\mathbb{Q}[G]} = \lim_n \mathbb{Q}[G]/I^n, I = \langle g-1 \,|\, g\in G\rangle.$ The nilpotent completion of $G$ is also defined by $G^{\mathrm{nil}}=\lim_n G/G^{(n)}, G^{(1)}=G, G^{(n+1)}= (G,G^{(n)}).$ These limit definitions induce two natural filtrations (both denoted by $F$) on Malcev and nilpotent completion. Also in both cases $\forall m,n, \,(F_n,F_m)\subset F_{n+m}$ and so there exists a natural structure of Lie algebras on the direct sum of graded quotients: $E\widehat{G} = \oplus F_n \widehat{G}/F_{n+1}\widehat{G}, EG^{\mathrm{nil}}= \oplus F_n G^{\mathrm{nil}}/F_{n+1}G^{\mathrm{nil}}$.

I have three questions on these concepts:

1- What is the relation between these two completion processes? I know (By theorems in Chapter 8 of Fresse's book: Homotopy of operads and Grothendieck-Teichmuller groups) in the case of finitely generated free groups, $E\widehat{G} \simeq EG^{\mathrm{nil}}\otimes \mathbb{Q}.$ Does this hold in general? If not, does there exist simple sufficient conditions for this to be true?

2-It's evident that $\widehat{G}^{(n)}\subset F_n(\widehat{G})$. Does there exist examples of $G$ for which this inclusion is strict for some $n$?

3-Is $\widehat{G}=\widehat{\widehat{G}}$ in general?

Thanks!