I should begin by apologizing for rambling on a bit. You've asked questions that are closely related to things I've been thinking about, but I don't know all the literature well, and I clearly haven't figured out how to say things concisely.

I will write $E_d$ for the operad (in spaces) of little $d$-disks, and $G_d$ ("$G$" for "Gerstenhaber") for what you're calling $\operatorname{Pois}^d$. (Because the word "$d$-Poisson algebra" appears at least in some papers by Cattaneo and collaborators, but at least the early ones get the sign wrong, and call by "$0$-Poisson" what should be called "$1$-Poisson".) Namely, $G_d$ is an operad in graded vector spaces generated by a binary "multiplication" in (homological) grading $0$ and a binary "bracket" in grading $d-1$ (and a zero-ary "unit" in grading $0$), such that the multiplication (and unit) is unital commutative, and such that the bracket is Lie up to shifting by $d-1$, and such that the Leibniz rule is satisfied. (In the correct sign conventions, saying that the bracket is "Lie after shifting" mean that it satisfies a Jacobi identity and is antisymmetric for odd $d$ and symmetric for even $d$.)

Begin with the operad $E_d$. The space of binary operations is a $d$-sphere. When $d\geq 1$, the $d$-sphere is nonempty, and its (integer!) homology is free on two generators, one in degree $0$ and the other in homological degree $d-1$. When $d=1$, please work in a setting where $2$ is invertible, and switch from the basis of points to the basis consisting of the average of the two points in the $0$-sphere and the difference. Let me call by "the bracket" any binary operation representing the degree-$(d-1)$ generator in homology (the fundamental class) and by "the multiplication" any binary operation representing the degree-$0$ class in homology. Let me also linearize by taking chains; so I will work with the operad of chain complexes $\mathrm C_\bullet(E_d)$, but write "$E_d$-algebra" for "$\mathrm C_\bullet E_d$-algebra".

Then the first observation is that the bracket in a fairly precise way measures the failure of the multiplication is be commutative. The second observation is that the bracket distributes over itself up to a contractible space, and is (anti)$^d$commutative, where "(anti)$^d$" means "anti" when $d$ is odd, and is empty when $d$ is even. So in fact if you throw away the multiplication and use only the bracket, there is a map to $\mathrm C_\bullet(E_d)$ from $\operatorname{Lie}[d-1]$, where by "$\operatorname{Lie}[d-1]$" I mean the Lie operad with the bracket shifted into homological degree $d-1$. And I mean that this map exists in some infinity sense. The point is that any $E_d$-algebra, if you remember only the action of the $d$-sphere, is a Lie algebra up to shifting by homological degree $(d-1)$, and up to replacing points with contractible spaces and all that. "Strongly homotopy" is a word that should now be bandied about.

Ok, so we can go a bit further. If you have a family of $E_d$-algebras over the (formal) disk $\operatorname{Spec}(k[\![\hbar]\!])$ that degenerates at the origin $\hbar=0$ to an $E_\infty$-algebra (recall that there are inclusions of operads $E_d \hookrightarrow E_{d+1}$, and the direct limit $E_\infty$ has a contracible space of $k$-ary operations for any $k\geq 0$, and so is "strongly homotopically" the commutative operad), then among other things you have a family of $\operatorname{Lie}[d-1]$-algebras that degenerates to the $0$-algebra at the origin. Then rescale the Lie bracket by $\hbar^{-1}$; now it need not vanish at $\hbar = 0$. The (homotopy-)associativity in the $E_d$-algebra implies a (homotopy-)Leibniz rule. So at the origin actually you have a (strongly homotopy) $G_d$-algebra.

This is all a long story to explain something short, which David Ben-Zvi has already alluded to: $G_d$ "is" the operad of "first order" $E_d$ algebras. Of course, I haven't at all shown that there aren't maybe further relations, or (which is the same thing!) further operations that survive (maybe after rescaling by $\hbar^{-1}$) at the limit. Essentially, the proof (that I know of) that there aren't any only applies in $d\geq 2$, and only in characteristic $0$: the celebrated fact that the homology $\mathrm H_\bullet(E_d)$ is on the nose $G_d$, as operad of chain complexes. I'm pretty sure, but don't know a reference, that this fails in non-zero characteristic. Point, is, when $d \geq 2$ it is clear that $\mathrm H_0(E_d)$ is (chains on) the commutative operad, because the other operations are all far away from $0$ in homological degree, and so there is a map $\mathrm H_\bullet(E_d) \leftarrow G_d$ in any characteristic when $d\geq 2$. It is somewhat remarkable that over $\mathbb Q$ this map is an iso; evidence that it is remarkable is that this fact comes after many wrong proofs.

Note that having a map $G_d \to \mathrm H_\bullet(E_d)$ is a far cry from having a map $G_d \to \mathrm C_\bullet(E_d)$. Sure, you can chose generators of homology and try to lift from $\mathrm H_\bullet$ to $\mathrm C_\bullet$, but there are spaces of ways to do this that are not contractible, and so you won't in general be able to make the lift be one of operads-up-to-strongly-homotopy. What I've tried to explain is a map from $G_d$ to the operad of "$E_d$-algebras over the formal disk that are $E_\infty$ at the point". (Recall: maps of operads go opposite to "forgetful" maps.)

Anyway, I'd like to end with a few comments on the too-celebrated "formality theorem". Formality is stronger than but implies the statement that every $G_d$-algebra is the first-order part of an $E_d$-algebra. Here's an example. Let me work with categories enriched over $\mathbb Q$. A *Casimir category* is a symmetric monoidal category equipped with a natural isomorphism between $\otimes$ and $\otimes\circ \operatorname{flip}$ satisfying something like Jacobi and Leibniz. Then formality with $d=2$ implies that every Casimir category is the first-order part of a braided monoidal category.

When $d=2$, Tamarkin pointed out that formality is the same as existence of something called "Drinfeld associators". $d=2$ is the number I understand the best, because Drinfeld associators are a bit older than the general formality theorem (to which I don't believe there is a proof over $\mathbb Q$; at least, Lambrechts and Volic, in their excellent paper on Kontsevich's proof of formality, say that for the (correct) unital versions of the operads, the trick that gets from Kontsevich's proof, which is over the periods, to something over $\mathbb Q$, doesn't work). Proofs of formality, i.e. Drinfeld associators, are the points of a scheme (or something like one — maybe I want to work more homotopically, and maybe I want projective limits, and ...). By "proofs of formality" I mean more or less "morphisms of operads". Anyway, this scheme (at least for $d=2$) is non-empty over $\mathbb Q$ (and for all $d$ it is nonempty over the periods), but it is never "contractible", whatever that means for this type of scheme-like object. So you really are making a choice when you invoke formality, and some choices are better than others. For example, some are more amenable to explicit computations, some satisfy extra symmetries, etc.. And there is no getting around this, because the groups of automorphisms of $G_d$ or $E_d$ are not contractible, and the space of isomorphisms is, of course, a bitorsor for these two groups.

Ok, a conclusion. I'm pretty sure the answer to your last question is "yes". But I'm also pretty sure that it hasn't been written down, really. (I would love to be told otherwise!) At least, I'm not aware of a paper that explains well the situation when $d\geq 3$. When $d=2$, there is plenty, including all the work by Etingof and collaborators on associators and Lie bialgebra (bi)quantization; these papers tend to be long, but understandable on about the fourth read. Some of it is well-reviewed by Bar-Natan, who is an excellent writer. Tamarkin's papers are short, and I find hard to read, but have a lot in them. I do highly recommend the papers by Severa on this stuff, and in particular his write-up of Tamarkin's work. A last part of the $d=2$ story is related to words like "Alekseev Torossian" and "Kashiwara Verne", and I don't understand any of that part.

isa passage to an associated graded (for the "Postnikov filtration" $F_k V= \tau_{\geq k} V$). The $d$-Poisson algebra $H_*(\Omega^d X)$ is the associated graded of the $E_d$-algebra $C_*(\Omega^d X)$. (Here by $E_d$ I mean the operad of chains on the little $d$-disks operad. It's a filtered operad in the manner above, $C_*(\Omega^d X)$ is then a filtered algebra over this operad $E_d$, etc. etc.) $\endgroup$