In this answer, the notion of $R$-immersion of ringed spaces was suggested. It means a topological embedding $f:X\to Y$ such that germ restriction on the structure sheaves is surjective. $$\mathcal O_{Y,fx}\twoheadrightarrow \mathcal O_{X,x}$$
I am trying to understand what this condition might mean for smooth manifolds. In this case the structure sheaves are the sheaves of smooth real functions. Let $f:X\to Y$ a smooth map. Precomposing with it induces a restriction map on germs $$C^\infty_{Y,fx}\to C^\infty_{X,x}.$$
Question 1. Surjectivity means a smooth germ on $X$ extends along $f$ to a smooth germ on $Y$? When does this hold? I don't see why it should hold for smooth immersions, for instance. (Or for smooth embeddings.)
Question 2. Suppose that instead of the sheaves $C^\infty$ we consider the tangent sheaves, i.e the sheaves of sections of the tangent bundles. When do we have surjectivity of germ restriction $$T_{Y,fx}\to T_{X,x}?$$ (This seems closer to smooth immersions.)
For question 1:
Note that if $\phi$ is a germ at $f(x)$, then $f^*\phi$ is a germ at $x$, and its differential at $x$ is $d_{f(x)}\phi \circ d_x f$. Thus for any $\phi$, the differential at $x$ of $f^*\phi$ must vanish on the kernel of $d_x f$.
So, if $f^*$ is surjective, $f$ must be an immersion at $x$.
Conversely, a well-known theorem states that if $f$ is immersive at $x$, then there are smooth coordinates around $x$ and $f(x)$ such that $f(y_1, \ldots, y_r)=(y_1, \ldots, y_r, 0, \ldots, 0)$. As a consequence, there exists a smooth $\pi : U \subset Y \rightarrow X$, defined on a neighborhood of $f(x)$, such that $\pi \circ f=Id$.
As a consequence, if $\phi$ is a germ around $x$, $\phi \circ \pi$ is a germ around $f(x)$ and $f^* \phi \circ \pi=\phi$, thus $f^*$ is surjective.
I am not sure about your second question, because I don’t understand what exactly your spaces are, and what your pull-back operator is.