"Standard reference" for $C_c^\infty(\mathbb R)$ is dense in $C_c(\mathbb R)$ $C_c^\infty(\mathbb R)$ is dense in $C_c(\mathbb R)$. This can be shown by mollification. This is a well-known, widely used fact. However, I wasn't able to find any book which I could point in a reference to. Is there any kind of "standard reference" with a readable proof?

I don't have a reference at hand. But one can prove this without too much trouble using the Weierstrass approximation theorem.
Suppose $f\in C_c$ with support contained in $[a,b].$ Let $\epsilon>0.$ Choose $a'<a$ and $b'>b.$  Then there exists a polynomial $p$ such that $|p-f|<\epsilon$ on  $[a',b'].$
Now there exists $g\in C^\infty_c(\mathbb R)$ with support in $[a',b']$ such that $0\le g\le 1$ everywhere, and $g=1$ on $[a,b].$ We then have $gp\in C^\infty_c(\mathbb R),$ and $|gp -f|<\epsilon$ everywhere. I'll leave the vefication of the last line to the reader; ask questions if you like.
