I'm not sure I can answer everything Adrien asked, but perhaps I can explain a little about part 6 in Tamarkin's proof, and about how the Drinfeld double appears in Tamarkin's story.
First, I'd like to explain a slightly different way to phrase Tamarkin's story. It's relies on Koszul duality, and throughout, there are some subtleties about completness, finiteness, etc. which always appear in Koszul duality. These can be dealt with, but I'll surpress the details.
The key thing we need to know is that any augmented $E_2$ algebra produces a Hopf algebra. (This is part 6 in Tamarkin's proof, as explained in the question). We can see this using Tanakian theory and Koszul duality. If $A$ is an augmented associative algebra, with a map $A \rightarrow k$ to the ground field, we can form a Koszul dual associative algebra
$$
A^! = \mathrm{Hom}_k(A,A).
$$
You can find (in good situations) an equivalence
$$
A^!\mbox{-}\mathrm{mod} = A\mbox{-}\mathrm{mod}.
$$
Actually, a precise version of this statement is pretty subtle; see e.g. Positselski's work. The coalgebra version of this statement (where we use comodules over $k \otimes_A k$) works better.
Let's assume for simplicity that $A^!$ has cohomology in degree $0$, so we don't have to worry about homotopical analogs of Hopf algebra structures.
Now, if $A$ is an $E_2$ algebra then left modules over $A$ form a monoidal category $A\mbox{-}\mathrm{mod}$ (this is explained in Lurie's higher algebra, for instance). Thus $A^!\mbox{-}\mathrm{mod}$ is a monoidal category, and you can check that the forgetul functor
$$
A^!\mbox{-}\mathrm{mod} \rightarrow \mathrm{dgVect}
$$
is monoidal. Thus by Tannakian theory $A^!$ has a Hopf algebra structure.
There's a similar story for $E_3$ algebras. Suppose $A$ is an $E_3$ algebra which is augmented as an $E_2$ algebra. Then $A\mbox{-}\mathrm{mod}$ has an $E_2$ (i.e. braided monoidal) structure, so that $A^!\mbox{-}\mathrm{mod}$ is a braided monoidal category such that the forgetful functor to vector spaces is monoidal. This implies, I believe, that $A^!$ has the structure of a quasi-triangular Hopf algebra.
Let $P_2$ be the homology operad of $E_2$. Formality of the $E_2$ operad gives an isomorphism between $E_2$ and $P_2$, and allows us to turn any $P_2$ algebra into an $E_2$ algebra. Lie bialgebras give rise to augmented $P_2$ algebras, as follows. If $\mathfrak{g}$ is a Lie bialgebra, then $C^\ast(\mathfrak{g})$ is a commutative algebra, with a Poisson bracket defined on generators $\mathfrak{g}^\vee$ by the Lie coalgebra structure on $\mathfrak{g}$.
This story (due to Tamarkin, of course) produces a Hopf algebra from any Lie bialgebra (subject to various caveats about completeness, etc. etc. which is why we find a formal quantization instead of an actual quantization)
The question asked about the Drinfeld double. I would guess that this works as follows. The Drinfeld double of a Hopf algebra is a quasi-triangular Hopf algebra. Quasi-triangular Hopf algebras are Koszul dual to $E_3$ algebras, and under Koszul duality, Drinfeld double should corresponds to $E_2$ Hochschild cohomology. By the higher analog of Deligne's conjecture, the $E_2$ Hochschild cohomology of an $E_2$ algebra is an $E_3$ algebra.
Some aspects of this story about $E_2$ algebras and Hopf are explained in detail in a paper I plan to put on the arxiv soon (which is closely related to Johnson-Freyd and Gwilliam's work). Of course, the picture is not due to me, but to Tarmarkin, but I don't think he showed that there's an equivalence of monoidal categories in this story.
Best Answer
There is a nice summary of the relationship between B infinity and G infinity in the first chapter of the book "Operads in Algebra, Topology and Physics" by Markl, Stasheff and Schnider. The short answer is G infinity is the minimal model for the homology of the little disks operad (the G operad). B infinity is an operad of operations on the Hochschild complex. Many of the proofs of Deligne's conjecture involve constructing a map between these two operads.