This wonderful and insightful Wikipedia page contains a lot of interesting facts about representations of diffeomorphism groups. However, there are no references.
In general, I'm curious if $Diff_k^p(\mathbb{R}^d)$ denotes the orientation-preserving $C^k$-diffeomorphism group of $\mathbb{R}^d$ which stabilizes $0$. They claim that $Diff^{\infty}_x(\mathbb{R}^d)/Diff^{1}_x(\mathbb{R}^d)$ can be identified with $GL(\mathbb{R}^d)$. Where can I find this fact? More interestingly, what can we say about $$ Diff^{\infty}_x(\mathbb{R}^d)/Diff^{k}_x(\mathbb{R}^d), $$ for $k\geq 2$ (besides it being finite-dimensional) and where can I find a nice book about these wonderful things?
I think the title of your question is slightly misleading, what you are looking at is jet groups rather than diffeomorphism groups. So these are finite dimensional Lie groups rather than the infinite dimensional diffeomorphism groups and they are independent of the manifold one considers. Of course, they do give rise to representations of the groups of diffeomorphisms fixing a point, but again these do not see the manifold.
In my opinion, the defintion that is chosen in the Wikipedia page you link is not the most transparent one. There is a general notion of $k$-jets of smooth maps between manifolds. In particular, for each $k$ and fixed $n$, you can consider $k$-jets at $0$ of smooth maps $\mathbb R^n\to\mathbb R^n$ which map $0$ to $0$. In there the jets of diffeomorphisms form an open subset and a group $G^k_n$ under compositions. For $k=1$, a 1-jet of such a smooth map is a linear map $\mathbb R^n\to\mathbb R^n$ (representing the derivative of the map at $0$) and this is the 1-jet of a diffeomorphism if and only if it is a linear isomorphism. Hence $G^1_n\cong GL(n,\mathbb R)$ and mapping a $k$-jet to the underlying 1-jet defines a surjective homomorphism $G^k_n\to G^1_n$, so each jet group is an extension of $GL(n,\mathbb R)$. To get to your language, you can send a diffeomorphisms of $\mathbb R^n$ that fixes $0$ to its $k$-jet, thus defining a surjective homomorphism $Diff_0(\mathbb R^n)\to G^k_n$ for each $k$. The kernel of this homomorphism is what you denote by $Diff^k_0(\mathbb R^n)$, since as mentioned in the comment by @Nate, these are diffeomorphisms that equal the identity to $k$th order.
Representations of the jet groups are closely related to natural vector bundles and thus to geometric objects on smooth manifolds. This is discussed in the book "Natural Operations in Differential Geometry" by Kolar, Michor and Slovak, that is availble online via Peter Michor's homepage here . But there is not as much explicit representation theory in that book.