@JochenWengenroth the result from Schwartz's book you quoted is a remark ("Remarque importante" in page 244-245) which seems to be only stated, not really proved. Is the proof elsewhere in the book (I just didn't see it), or is there another reference with the proof? Thanks in advance.

This is consistent with seeing $f\in\text{Diff}(M)$ as a smooth section of the trivial bundle $\text{pr}_1:M\times M\ni(p_1,p_2)\mapsto p_1\in M$ once we identify $f$ with its graph as before. That's where the nonlinearity of the (first order) differential operator $(f,\omega)\mapsto f^*\omega$ comes in, for the differential operator $\omega\mapsto f^*\omega$ by itself is linear and of order zero. Likewise, one must see $(f,\omega)$ as smooth sections of the Whitney sum of $\text{pr}_1$ and $\tilde{\pi}$.

Recall, though, that the notion of differential operators you're recalling from Hamilton doesn't strictly apply here because it actually defines linear differential operators. A way to see the pullback as a differential operator is to see the pullbacks of $\wedge^n T^∗\! M$ by all $f\in\text{Diff}(M)$ as sub-bundles of the (fiber) bundle $\tilde{\pi}:M\times\wedge^n T^*\! M\rightarrow M$, the bundle projection being $(p,q)\mapsto \tilde{\pi}(p,q)=p$.

You must consider the pullback $f^∗\omega$ of $\omega\in\Omega^n(M)=\Gamma(\wedge^n T^*\! M)$ as a smooth section of the pullback vector bundle $f^*\wedge^n T^*\! M$ (the identification being through the graph of $f^*\omega$ as usual). Once you do that, the pullback map $\omega\mapsto f^*\omega$ becomes a proper differential operator (mapping smooth sections of $\wedge^n T^*\! M$ to smooth sections of $f^*\wedge^n T^*\! M$).

What about $2p$-vector fields with $p>1$ then?

This MO question might be of interest: mathoverflow.net/q/155989/11211 - In that regard, it can be shown that a 2-vector field $X=X_1\wedge X_2$ defines an integrable 2-dimensional distribution iff its Schouten-Nijenhuis bracket with itself vanishes everywhere. This also provides a characterization of when a 2-vector field is a Poisson tensor, check e.g. P. W. Michor, Remarks on the Schouten-Nijenhuis bracket, Rend. Circ. Mat. Palermo Suppl. 16 (1987), pp. 207-215, dml.cz/dmlcz/701423 . I don't know of a similar geometric interpretation of $p$-vector fields for $p>2$.

Another thing that follows from the fact that $f$ is (measurable, ) compactly supported and bounded is that $\hat{f}$ is a polynomially bounded real analytic function, by the Paley-Wiener-Schwartz theorem. Your final hypothesis just strengthens the polynomial bound on $\hat{f}$. However, that additional regularity on $\hat{f}$ doesn't seem to bring anything new to the considerations in my previous comment.

I suppose that $f$ is at least measurable (hence $L^1$ and identified with a tempered distribution in $\mathbb{R}^2$). The validity of the first limit follows (from the Riemann-Lebesgue lemma) if the (distributional) Laplacian $\Delta f$ of $f$ is in $L^1$ as well. However, since the Fourier transform maps $L^1$ into but not onto $C_0$, it may not be true that $\Delta f\in L^1$ with your hypothesis alone. Also, unfortunately you are "on the wrong side" of the Sobolev embedding theorem, so the latter result most likely won't help you either...

You may want to check out mathoverflow.net/a/157219/11211 and the discussion in the comments... TL&DR: Gel'fand-Vilenkin's book, "Generalized Functions - Volume IV", Chapter IV, pp. 303ff..