A couple of quick comments.

Firstly, I'd advise you to think about what you know about Q1 in the case $G= {\mathbb Z}^d$ (or even just the case $G={\mathbb Z}$) -- because in such settings you can think of the group C*-algebra as the algebra of continuous functions on the Pontryagin dual. In that case, your question becomes one about examples of continuous functions which don't have absolutely summable Fourier coefficients.

Secondly, I think your definition of C is not quite right -- don't you need to take the ideal generated by all the commutators, not just the set of all commutators? (It feels like what you want to be asking about is the centre of $C_r^*(G)$ -- I think Losert has a paper on this -- but perhaps I have misunderstood.)

A couple of quick comments.

Firstly, I'd advise you to think about what you know about Q1 in the case $G= {\mathbb Z}^d$ (or even just the case $G={\mathbb Z}$) -- because in such settings you can think of the group C*-algebra as the algebra of continuous functions on the Pontryagin dual. In that case, your question becomes one about examples of continuous functions which don't have absolutely summable Fourier coefficients.

Secondly, I think your definition of C is not quite right -- don't you need to take the ideal generated by all the commutators, not just the set of all commutators? (It feels like what you want to be asking about is the centre of $C_r^*(G)$ -- I think Losert has a paper on this -- but perhaps I have misunderstood.)

Added 28-05-2010: The paper of Losert which I was dimly recalling is

[ Zbl 1088.22003] On the center of group C * -algebras. (English)

V. Losert, J. Reine Angew. Math. 554, 105-138 (2003).