Q1 seems to be related to symmetry of $\ell^1(G)$ (a Banach $^*$-algebra $A$ is symmetric if, for every $a\in A$, the spectrum of $a^*a$ is contained in $\mathbb{R}^+$). If $\ell^1(G)$ is not symmetric, then for an element $a^*a$ with non-positive $\ell^1$-spectrum, since the $C^*$-spectrum is clearly positive, the resolvent of $a^*a$ will define elements in $C^*_r(G)\backslash\ell^1(G)$. For examples of groups with non-symmetric algebras, see papers by J. Boidol, H. Leptin and D. Poguntke from the 1980's.