Let $\Pi$ be a Poisson structure on a manifold $M$. Then we can define a differential $d$ on the complex $\Lambda^{\bullet}M$ $$ C^{\infty}M \to TM \to...\Lambda^kTM \to... $$ in the following way: $$ d = [\Pi,\cdot], $$ where $[\cdot,\cdot]$ is the Schouten-Nijenhuis bracket. The cohomology of this complex is called Poisson cohomology.

I have several question about this construction:

Suppose $M=\mathbb R^{2d+s}$ with a Poisson structure $\sum\limits_{i=1}^{d}\frac{\partial}{\partial q_i}\wedge \frac{\partial}{\partial p_i}$. Is it true that Poisson cohomology of $M$ are zero?

Is this construction functorial? I mean, let $f\colon M\to N$ be a poisson map, then do we have a map of complexes $f_*\colon \Lambda^{\bullet}M \to \Lambda^{\bullet}N$?

Let $F\colon M\times I \to N$ be a homotopy such that for each $t$ $F_t$ is a Poisson map. Is it true that $F_0$ and $F_1$ induce the same map from Poisson cohomology of $M$ to Poisson cohomology of $N$?