To add some insights into my discussion. I found out some stuff that might be interesting according to matrix perturbation theory.
So instead of asking as in the 1st question: given prescribed eigenvalues what is the minimum $c_{max}$? I tried to look into the problem of what is the possible eigenvalue densities given I have a prescribed $c_{max}$.
It goes like this... I compare my correlation matrix with the identity matrix. The diagonal elements of both matrices are 1's, therefore the perturbation is in the off-diagonal elements. The Weyl-Lidskii theorem states that:
$|\lambda_i-1|\leq ||\mathbf{C}-\mathbf{I}||_2$
where I write $|\lambda_i-1|$ because the eigenvalues of the identity matrix $(\mathbf{I})$ are $1$. Therefore I can give a bound on the maximum variation of my correlation matrix eigenvalues.
$1-M(M-1)c_{max} \leq \lambda_i\leq 1+M(M-1)c_{max}$ for any $i$
However I still do not know what eigenvalue densities satisfying the previous equation are possible.