Since my question is related to sign convention, I want to define everything from the very beginning. $T\_{poly}^k(M)=\Gamma(\wedge^{k+1} TM)$ are the multi vector fields with shifted degree and with the Schouten bracket uniquely determined by 
\begin{align\*}
[\![f,g]\!] =0\ \text{, } [\![X,f]\!]=X(f) \text{ and } [\![X,Y]\!]=[X,Y] 
\end{align\*}
for $X,Y\in T\_{poly}^0(M)$ and $f,g\in T\_{poly}^{-1}(M)$. Hence 
$(T\_{poly}^\bullet(M),d=0, [\![-,-]\!] )$ is a DGLA.


Let us denote by 
\begin{align\*}
D\_{poly}^k=
\begin{cases}
\mathrm{DiffOp}(C^{\infty}(M)^{\otimes(k+1)},C^{\infty}(M)),\text{ for }k\geq 0 \\
C^\infty(M), \text{ for } k=-1\\
0, \text{ else } 
\end{cases}
\end{align\*}
 the multi differential operators with shifted degree by $1$. 
For $f\_i\in C^{\infty}(M)$, $\phi\in D\_{\mathrm{poly}}^k(M) $ and $\psi\in D\_{\mathrm{poly}}^l(M)$, we define
\begin{align\*}
 (\phi\circ\_i \psi)(f\_0,\dots,f\_{k+l})=\phi(f\_0,\dots, \psi(f\_i,\dots, f\_{i+l}),\dots,f\_{k+l})
\end{align\*}
for $k,l\geq 0$ and 
\begin{align\*}
 (\phi\circ\_i \psi)(f\_0,\dots,f\_{k+l})=\phi(f\_0,\dots,f\_{i-1}, \psi,f\_{i}, \dots,f\_{k})
\end{align\*}
for $k\geq 0$ and $l=-1$.
 The Gerstenhaber product is defined by 
 \begin{align\*}
 \phi\circ \psi=\sum\_{i=0}^{\deg \phi}(-1)^{i\deg\psi} \phi\circ\_i\psi
 \end{align\*}
and finally 
 \begin{align\*}
 [\phi,\psi]=\phi\circ\psi -(-1)^{\deg\phi\deg\psi}\psi\circ\phi.
 \end{align\*} 
The point-wise multiplication of functions $m\_0\in D\_{poly}^1(M)$ fulfills $[m\_0,m\_0]=0$ and hence $\delta(x)=[m\_0,x]$ is a differential, which makes 
$(D\_{poly}^\bullet(M),\delta,[-,-])$ into a DGLA. As far as I got, Kontesevich proved the existence of an $L\_\infty$-morphism $F$ between $T^\bullet\_{poly}(M)$ and $D^\bullet\_{poly}(M)$ uniquely determined by is Taylor coefficients 
\begin{align\*}
F\_k\colon \wedge^k T^\bullet\_{poly}(M)[1]\to D\_{poly}^\bullet(M)[1-k]
\end{align\*}
where 
\begin{align\*}
F\_1(X\_1\cdots X\_k)(f\_1,\dots,f\_k)=
\frac{1}{k!}\sum\_{\sigma\in S\_k}X\_{\sigma(1)}(f\_1)\cdots X\_{\sigma(k)}(f\_k)
\end{align\*}
is the Hochschild-Kostant-Rosenberg map which is a chain map. Since $F$ is a $L\_\infty$-morphism, we know that for a Poisson structure $\pi\in T^1\_{poly}(M)$ the element $\hbar\pi$ is a Maurer-Cartan element in $T\_{poly}^\bullet(M)[[\hbar]]$ and so 
\begin{align\*}
\hbar m\_\star:= \sum\_{k=1}^\infty 
\frac{\hbar^k}{k!}F\_k(\pi\wedge\dots \wedge \pi) 
\end{align\*}
is a Maurer-Cartan element and hence $m\_0+\hbar m\_\star$ is a star product. Moreover, we have then two new DGLA's 
\begin{align\*}
(T^\bullet\_{poly}(M)[[\hbar]], d\_\pi=[\![\hbar\pi,-]\!], [\![-,-]\!]) 
\text{ and }
(D^\bullet\_{poly}(M)[[\hbar]], \delta\_\star=[m\_0+\hbar m\_\star,-],[-,-])
\end{align\*} 
and a $L\_\infty$-morphism $F^\pi$ uniquely determined by 
\begin{align\*}
F^\pi\_l (X\_1\wedge\dots\wedge X\_l)=\sum\_{i=0}^{\infty} \frac{\hbar^i}{i!}F\_{i+l}(\pi\wedge\dots\wedge\pi\wedge X\_1\wedge\dots\wedge X\_l)
\end{align\*}
Since $F^\pi$ is an $L\_\infty$-morphism we know again that $F^\pi\_1$ is a chain map. Let us denote by $X\_f=[\![\pi,f]\!]$, then we have 
\begin{align\*}
\hbar \mathcal{L}\_{X\_f}&= \hbar F\_1(X\_f)=F\_1(d\_\pi(f))\\&
=\sum\_{i=0}^\infty \frac{\hbar^i}{i!} F\_{i+1}(\pi\wedge\dots\wedge\pi\wedge d\_\pi(f))\mod\hbar^2\\&
=F\_1^\pi(d\_\pi(f))\mod\hbar^2\\&
=\delta\_\star F\_1^\pi(f) \mod\hbar^2\\&
=[f,-]\_\star \mod\hbar^2\\&
=\hbar F\_1(\pi)(f,-)\mod\hbar^2
\end{align\*}
where I used that for a function $f$ the differential $d\_\star $ is just the insertion in the first argument of the (deformed) product minus the insertion in the second argument. But now we have that $F\_1(\pi)(f,-)=\mathcal{L}\_{\pi^\sharp(df)}=\mathcal{L}\_{-[\![\pi,f]\!]}=-\mathcal{L}\_{X\_f}$, which clearly contradicts the left hand side of the equation.


Now, my questions:


(1) Are my computations with the defined DGLA structures correct?


(2) If no, where exactly is the mistake?


(3) If yes, which is the right DGLA structure on $D\_{poly}^\bullet(M)$, such that the $F\_k$'s are an $L\_\infty$-morphism. (I am assuming that $D\_{poly}^\bullet(M)$ has a wrong sign convention, since in the literature the Schouten bracket is usually defined the same ways, but in the definition of the structures on the Hochschild complex, there are many different conventions.)



Kontsevich Formality sign convention
Welcome to mathoverflow! 


There is actually a whole paper (in French) about choices of signs for Kontsevich formality: <https://arxiv.org/pdf/math/0003003.pdf>


For instance, they define the Hochschild coboundary operator as $\delta(x)=-[m\_0,x]$ (see also their Remark on page 20). 


I hope this will help. 


