When is $C_c^\infty(\mathbb R^d \setminus \{ 0 \})$ dense in $C_c^\infty(\mathbb R^d)$? Let $d\in \mathbb N$. Under what constraints for $d$ and $p$  is $C_c^\infty(\mathbb R^d \setminus \{ 0 \})$ dense in $C_c^\infty(\mathbb R^d)$ with respect to the $L^p(\mathbb R^d)$-norm?
Let $b_n$ be a sequence of smooth functions $b_n: \mathbb{R} \to [0,1]$ such that $b_n(x) = 1$ for $|x|&gt; { 2\over n}$ and $b_n(x) = 0$ for $|x|&lt; {1 \over n}$.

If $f \in C_c^\infty(\mathbb R^d)$, then $b_n\cdot f \in C_c^\infty(\mathbb R^d \setminus \{0\})$ and for $p&lt; \infty$ we have $\|f-b_n \cdot f\|_p \to 0$.

If $f \in C_c^\infty(\mathbb R^d)$ such that $f(0) = 1$, then we must have
$\|f-g\|_\infty \ge 1$ for any $g \in C_c^\infty(\mathbb R^d \setminus \{0\})$.
I'm not sure that your question makes sense, since the first set is not a subset of the second. I don't think you can embed it in any reasonable way, since it is not always possible to extend a continuous function from the first set to the other. I'd guess for $d=1$, something like $sin(1/x)$ smoothed out to some compact domain should be an example.

Edit: I guess setting $f(0)=0$ should work as an embedding, since any compact set in $\mathbb{R}^d$ should have empty intersection with some ball around $0$.