A few questions about Kontsevich formality [K] refers to Kontsevich's paper "Deformation quantization of Poisson manifolds, I".
Background
Let $X$ be a smooth affine variety (over $\mathbb{C}$ or maybe a field of characteristic zero) or resp. a smooth (compact?) real manifold. Let $A = \Gamma(X; \mathcal{O}_X)$ or resp. $C^\infty(X)$.
Denote the dg Lie algebra of polyvector fields on $X$ (with Schouten-Nijenhuis bracket and zero differential) by $T$. Denote the dg Lie algebra of the shifted Hochschild cochain complex of $A$ (with Gerstenhaber bracket and Hochschild differential) by $D$.
Then the Hochschild-Konstant-Rosenberg theorem states that there is a quasi-isomorphism of dg vector spaces from $T$ to $D$. However, the HKR map is not a map of dg Lie algebras. It is not a map of dg algebras, either (where the multiplication on $T$ is given by the wedge product and the multiplication on $D$ is given by the cup product of Hochschild cochains).
I believe "Kontsevich formality" refers to the statement that, while the HKR map is not a quasi-isomorphism --- or even a morphism --- of dg Lie algebras, there is an $L_\infty$ quasi-isomorphism $U$ from $T$ to $D$, and therefore $D$ is in fact formal as a dg Lie algebra.
The first "Taylor coefficient" of the $L_\infty$ morphism $U$ is precisely the HKR map (see section 4.6.2 of [K]).
Moreover, this quasi-isomorphism $U$ is compatible with the dg algebra structures on $T$ and $D$ (see section 8.2 of [K]), and it yields a "corrected HKR map" which is a dg algebra quasi-isomorphism. The "correction" comes from the square root of the $\hat{A}$ class of $X$. See this previous MO question.
Questions
(0) Are all of my statements above correct?
(1) In what way is the $L_\infty$ morphism $U$ compatible with the dg algebra structures? I don't understand what this means.
(2) When $X$ is a smooth (compact?) real manifold, I think that all of the statements above are proved in [K]. When $X$ is a smooth affine variety, I think that the statements should all still be true. Where can I find proofs?
(3) Moreover, the last section of [K] suggests that the statements are all still true when $X$ is a smooth possibly non-affine variety. For a general smooth variety, though, instead of taking the Hochschild cochain complex of $A = \Gamma(X;\mathcal{O}_X)$, presumably we should take the Hochschild cochain complex of the (dg?) derived category of $X$. Is this correct? If so, where can I find proofs?
In the second-to-last sentence of [K], Kontsevich seems to claim that the statements for varieties are corollaries of the statements for real manifolds, but I don't see how this can possibly be true. In the last sentence of the paper, he says that he will prove these statements "in the next paper", but I'm not sure which paper "the next paper" is, nor am I even sure that it exists, since "Deformation quantization of Poisson manifolds, II" doesn't exist.
P.S. I am not sure how to tag this question. Feel free to tag it as you wish.
 A: Hi Kevin, even if the question is answered I would like to add a few remarks. 
(0) the claim that 

this quasi-isomorphism $U$ is
  compatible with the dg algebra
  structures on $T$ and $D$

is not exactly true. It is compatible only on tangent cohomology. 
(1) I agree with Daniel and James that there exists a $G_\infty$-quasi-isomorphism between $T$ and $D$ (this implies compatibility at the level of tangent cohomology, and is strictly stronger). But until recently this was not known if Kontsevich's $L_\infty$-quasi-isomorphism can be upgraded to a $G_\infty$-one. A recent paper of Willwacher solves (in a positive way) this question (EDIT: Willwacher makes the comparison with Tamarkin's $G_\infty$-quasi-isomorphism in Section 10). 
(2) the proof for smooth affine varieties is essentially the same as the one for smooth differentiable manifolds. Both rely 


*

*either on the exitence of a connection in the tangent bundle (see the papers of Dolgushev, e.g. this one). 

*or (equivalently) on acyclicity of sheaves of sections of bundles (see e.g. my paper with Michel Van den Bergh). 
(3) references are 


*

*Yekutieli's paper and Van den Bergh'sone for smooth algebraic varieties. 

*my paper with Dolgushev and Halbout for complex manifolds. 

*the above paper with Van den Bergh for a uniform approach to smooth, complex and algebraic settings (using Lie algebroids). 

*Dolgushev-Tamarkin-Tsygan paper for the $G_\infty$ version (see also another paper with Van den Bergh). 

For a general smooth variety, though,
  instead of taking the Hochschild
  cochain complex of $A=\Gamma(X;\mathcal O_X)$,
  presumably we should take the
  Hochschild cochain complex of the
  (dg?) derived category of $X$. Is this
  correct?

One could do this, but one instead works on the sheaf level. Consider $T$ and $D$ as sheaves and prove that they are $L_\infty$- (or $G_\infty$-)quasi-isomorphic as sheaves of DG Lie (or $G_\infty$)-algebras. 
(3+$\epsilon$) "Deformation quantization of Poisson manifolds, II" does not exist, but there is "Deformation quantization of algebraic varieties" (quite sketchy). You might also be interested by the very inspiring paper "Operads and motives in deformation quantization". 
A: To (1): Daniel is right, there is a map of homotopy Gerstenhaber algebras between the two algebras.  However the full story is quite complicated and to show that the hochschild cochains form a homotopy Gerstenhaber algebra is hard, it's known as the Deligne conjecture.  I don't know the details of the proof.
Recall that a Poisson algebra is a commutative algebra with a Lie bracket and these two products satisfy a Leibniz identity.  A Gerstenhaber algebra is a bit like a Poisson algebra, except the Lie bracket is of degree 1 not 0.  The bracket satisfies a graded Leibniz identity wrt to the commutative algebra structure.
The formality morphism as homotopy Gerstenhaber algebras restricts to a formality morphism as homotopy Lie algebras and to a formality morphism as homotopy commutative algebras.
In my view the simplest proof of the formality of the Hochschild cochains of a nice enough algebra as a homotopy Gerstenhaber algebra is contained in
http://arxiv.org/abs/math.KT/0605141
A: Here are answers to questions 2, 3 and 1 (in this order). 
2) Let $X = \operatorname{Spec} A$ be a smooth affine variety over a field $K$ of characteristic $0$. Then there is a canonical bijection between Poisson deformations of $A$ and associative deformations of $A$ (all up to gauge equivalence). This was first proved (in slightly greater generality, yet assuming $K$ contains the real numbers) in my paper

Deformation Quantization in Algebraic Geometry, Amnon Yekutieli, 
  Advances in Mathematics 198 (2005), 383-432. Erratum: Advances in Mathematics  217 (2008), 2897-2906.

The condition about real numbers is not essential for the proof; all that is needed is a Formality Theorem for the power series ring 
$K[[t_1, \ldots, t_n]]$ satifying the invariance properties of Kontsevich. 
It should be noted that the proof above is of a global nature. A local argument can only be made on an affine scheme $X$ that admits etale coordinates (namely an etale map to $\mathbf{A}^n_K$). Bigger affine schemes (that do not admit such coordinates) require a gluing argument, based on formal geometry, and the vanishing of some cohomological obstructions.  
There are other later proofs of this same result (listed in another answer). I am only familiar with the proof by Van den Bergh (which is a variation on my proof).   
3) If $X$ is an arbitrary smooth scheme (over a field $K$ of characteristic $0$), then quantization is possible, but only in a stacky sense. This was suggested by Kontsevich in his paper 

M. Kontsevich, Deformation quantization of algebraic varieties, Lett. Math. Phys. 56 (2001), no. 3, 271-294.

The precise statement, and its proof, are in the paper (still not published!) 

[TDQ] Amnon Yekutieli, Twisted Deformation Quantization of Algebraic Varieties, http://arxiv.org/abs/0905.0488 .

In this more general situation one has to consider twisted (i.e. stacky) Poisson deformations of the sheaf $\mathcal{O}_X$, and twisted associative deformations of the sheaf $\mathcal{O}_X$. 
A twisted associative deformation $\mathcal{A}$ is a refined sort of stack of algebroids. The category of coherent left $\mathcal{A}$-modules is a deformation (as a stack of abelian categories, in the sense studied by Lowen and Van den Bergh) of the category of coherent $\mathcal{O}_X$-modules. This also goes in reverse: $\mathcal{A}$ can be recovered from its module category (basically by Morita theory). 
A twisted Poisson deformation is a new kind of algebro-geometric object. There is no Morita theory connected to it. 
Theorem 0.10.1 of [TDQ] states that there is a canonical bijection, called the twisted quantization map, between twisted Poisson deformations of $\mathcal{O}_X$  and twisted associative deformations of $\mathcal{O}_X$ (up to twisted gauge equivalence). 
There are lecture notes on this work: 
http://www.math.bgu.ac.il/~amyekut/lectures/twisted-defs-2013/notes.pdf 
and a survey article: 
http://www.math.bgu.ac.il/~amyekut/publications/tw-defs-surv/tw-defs-surv.html
The main paper [TDQ] relies on several auxiliary papers, including:

Central Extensions of Gerbes, Amnon Yekutieli, 
  Advances in Mathematics 225, Issue 1 (2010), 445-486.
MC Elements in Pronilpotent DG Lie Algebras,
  Amnon Yekutieli, J. Pure Appl. Algebra 216 (2012), 2338–2360
Deformations of Affine Varieties and the Deligne Crossed Groupoid,
  Amnon Yekutieli, Journal of Algebra 382 (2013), 115–143.
Combinatorial Descent Data for Gerbes,
  Amnon Yekutieli, to appear in J. Noncommutative Geometry. Eprint arXiv:1109.1919 at http://arxiv.org. 

There are several intriguing open questions in this subject. For instance Question 0.10.2 of [TDQ] regarding twisted deformation quantization of Calabi-Yau surfaces. 
1) The compatibility of the Kontsevich formality morphism with the associative DG algebra structures on polyvector fields and polydifferential operators is made precise and proved in the paper 

Hochschild cohomology and Atiyah classes,
  Damien Calaque and Michel Van den Bergh,
  Advances in Mathematics 224 (2010) 1839–1889. 

As I recall, they prove that the universal formality morphism is (in additions to being L_infty) an A_2 morphism (namely A_infty truncated at level 2). This is enough to deduce the formula for twisting the HKR morphism (by the square root of the Todd class) in order to respect the multiplications.
A: Its only a reference, and I don't feel competent to give a summary. I think the answer to question 1 can be found in http://www.math.jussieu.fr/~keller/publ/emalca.pdf . At least it mentions the analogy to the Duflo isomorphism, which is similiar to what you are asking about. It also takes a map of vector spaces, does some magic and it ends up being a map of algebras.  
A: I would guess that the the proper statement to understand (1) which mixes the Lie structure and the algebra structure is that this map is some sort of map of homotopy Gerstenhaber algebras. I don't really know this stuff(or anything else), but my impression based upon work of Fred Cohen is that the precise statement is that this should be a map of modules over the homology of the little disc operad E2, which I guess acts on the Hochschild cochains by the proof of the Deligne Conjecture. 
A: Theorem 11 in Section 4.4 of this survey by Dolgushev, Tamarkin, and Tsygan answers my question (2).
Theorem 12 kind of addresses my question (3), but the approach there seems to be different from the approach that I am imagining. I am more interested in the Hochschild complex of the derived category. However, I would not be surprised if the Hochschild complex of the derived category of the variety is related to the "sheaf of Hochschild complexes" on the variety, probably the former is the global sections of the latter?
