The value at $s=0$ or $s=1$ is known by the Class number formula. Here are some answers for the first derivative: see this http://mathoverflow.net/questions/87873/dedekind-zeta-function-behaviour-at-1 for the value at $s=1$. Since for abelian $L$ functions, the functional equation is very wellknown it doesn't matter whether you look at $s=0$ or $s=1$. After I have studied the references, I was ready to believe that not much is known at least for the first derivative beyond estimates.
No, see this http://mathoverflow.net/questions/87873/dedekind-zeta-function-behaviour-at-1 for the value at $s=1$. Since for abelian $L$ functions, the functional equation is very wellknown it doesn't matter whether you look at $s=0$ or $s=1$.