This is a follow-up to a previous question, where it was concluded that at any point of Euclidean space, the "infinitesimal isometry group" is $O(n) \times \mathbb{R}^n$ and the "pointed infinitesimal isometry group" is $O(n)$ (precise definitions of each are to follow).
Questions (working from most specific to most general examples):
Are the "infinitesimal (pointed) isometry groups" at each point of the 2-sphere $\mathbb{S}^2$ the same as those for the Euclidean plane? (They should be equal at each point since $\mathbb{S}^2$ is homogeneous.) Or are they different because of the sphere's positive curvature?
For an arbitrary Riemannian $2$-dimensional manifold, are the "infinitesimal (pointed) isometry groups" at each point functions of the Gaussian curvature? Is the Gaussian curvature insufficient information to predict them? Or are they totally independent of it?
For an arbitrary $n$-dimensional Riemannian manifold $M$, is it true that, at any point $p \in M$, one has that the "infinitesimal isometry group" is $O(n) \times \mathbb{R}^n$ and that the "pointed infinitesimal isometry group" is $O(n)$? I.e. does the curvature of $M$ at $p$ affect these groups?
Note: An answer to any of the above three questions will suffice for an answer. The first question seems like the easiest and the one I am trying to solve right now, but ultimately the final level of generality I am interested in is the third question. Again, please answer whichever you want.
(If the answer to any of the above questions turns out to be false, then these groups should not be called "infinitesimal isometry groups", since Riemannian manifolds are supposed to be "infinitesimally Euclidean". Hence the quotation marks around the terms.)
Definitions: Given an arbitrary metric space $(X,d)$ (not necessarily a Riemannian manifold),
A map $f: X \to X$ is called a local isometry at $x \in X$ if $x$ has two (possibly equal) neighborhoods $U_x$ and $V_x$, such that (the restriction of) $f$ maps $U_x$ isometrically onto $V_x$.
A map $f: X \to X$ is called a pointed local isometry at $x \in X$ if it is a local isometry at $x \in X$ and if $f(x)=x$, i.e. if $f$ fixes $x$.
The infinitesimal isometry group at $x \in X$ is the set of all germs of local isometries at $x$.
The pointed infinitesimal isometry group at $x \in X$ is the same as the above, except replacing "local isometries at $x$" with "pointed local isometries at $x$".
It was shown in this previous question that both of these sets are actually groups, and that for Euclidean space the former is the Euclidean group and the latter is the orthogonal group, i.e. the same as the "global (pointed) isometry groups".