Two questions about convolution

Question

Good day to everyone, I have the following two questions about convolutions of smooth functions with compact suppport ($C_c^k(\mathbb{R}^d)$ or $C_c^\infty(\mathbb{R^d})$).

1. If $f$ from $C_c^m(\mathbb{R}^d))$ and $g$ from $C_c^n(\mathbb{R^d})$, what can we say about $f*g$? I am quite sure that $f*g\in C_c^{\max (m,n)}(\mathbb{R}^d)$, but can it be more than this? Can it be that always $f*g\in C_c^ {m+n}(\mathbb{R}^d)$?
2. Is the set $\{f*g: f,g \in C_c^\infty(\mathbb{R}^d) \}$ dense in $L^1$? I suppose so but not sure. Thanks for Your help.