A reasonable conjecture is that 

$$ \operatorname{Germain}(x) = 2 C \int_{2^x} \frac{dx}{\log^2 x} + O( x^{1-\delta} )$$ for some $\delta >0$. This is the Hardy-Littlewood conjecture with power-saving error term.

A function-field analogue of this conjecture follows from the methods of my paper [On the Chowla and twin primes conjectures over $\mathbb F_q[t]$][1] with Mark Shusterman.

A good way to test the validity of these conjectures is to see how they relate to this asymptotic.  For example, Conjecture 1 follows as long as $x$ and $y$ are not too far apart.

The integral $\int_{2^x} \frac{dx}{\log^2 x}$ has an asymptotic expansion $$x /(\log x)^2 + 2x / (\log x)^3 + 6 x / (\log x)^4+  \dots= x / (\log^2 x - 2 \log x + O(1))$$ and the power savings error term is, once we put it on the denominator, smaller than that $O(1)$, thus it is reasonable to conjecture that

$$\operatorname{Germain}(x) = 2 C \frac{x}{ \log^2 x - 2 \log x + O(1) } $$

and one could even look for numerical evidence about how large this $O(1)$ should be.

So indeed Conjecture 2 should be wrong but a modified version with the $-2 \log x$ could well be correct. 


  [1]: https://arxiv.org/abs/1808.04001