The distribution of values of $L(1,\chi_d)$ as $d$ varies over fundamental discriminants has been extensively studied.  For example, see this paper of [Granville and Soundararajan][1] which gives uniform such results (and discusses other references and history).  The main result there shows that $L(1,\chi_d)$ is distributed like a random Euler product $L(1,X)= \prod_p (1-X(p)/p)^{-1}$ where $X(p)$ for primes $p$ denote independent random variables with $X(p)=1$ with probability $p/(2(p+1))$, $-1$ with probability $p/(2(p+1))$ and $0$ with probability $1/(p+1)$.   From this the last assertion you make about fundamental discriminants follows: you want to compute (for large $D$)
$$ 
\exp\Big( \frac{1}{|\{|d|\le D\}|} \sum_{|d|\le D} \log (1,\chi_d) \Big)
\sim \exp\Big({\Bbb E} (\log L(1,X)) \Big) = \prod_p \Big(\frac{p^2}{p^2-1}\Big)^{\frac{p}{2p+2}}.
$$
Small modifications to the same techniques would allow you to study the family of prime discriminants that you mentioned -- the only difference is in adjusting the probabilistic model to reflect the fact that very few prime discriminants will be divisible a given prime $p$ (as opposed to all fundamental discriminants this proportion is $1/(p+1)$). 

[1]: http://arxiv.org/abs/math/0206031