Is $C^{\infty}_c(R^n) $ dense in $C_0(R^n)$? If $C^{\infty}_c(R^n) $ is dense in $L^p(R^n)$ then is it dense in $C_0(R^n)$?

To answer the question if $C_c^\infty(\mathbb R^n)$ is dense in $C_0(\mathbb R^n)$. Yes, it is. Either one uses cut-off functions as in the other answer or one can just use the Stone-Weierstrass Theorem (actually the version for $C_0$-spaces): 
So notice that $C_c^\infty(\mathbb R^n)$ is clearly a subalgebra of $C_0(\mathbb R^n)$. Moreover, it can be seen easily by considering bump-functions that $C_c^\infty(\mathbb R^n)$ seperates the points, i.e., for all $x, y \in \mathbb R^n$, $x \neq y$ there is $f \in C_c^\infty(\mathbb R^n)$ such that $f(x) \neq f(y)$. Finally, by considering bump-functions one see that $C_c^\infty(\mathbb R^n)$ vanishes nowhere, which means that for each $x \in \mathbb R^n$ there is $f \in C_c^\infty(\mathbb R^n)$ such that $f(x) \neq 0$. 
Hence, the Stone-Weierstrass Theorem yields the density of $C_c^\infty(\mathbb R^n)$. I hope that answers your question. 
