Question 1. Yes, and this follows from the results of Ch. VIII in Weil: Basic Number Theory. See especially Cor. 2 of Prop. 4 in that chapter.
Question 2. In general, the exponent of $p$ in $d$ equals $\lceil g_p/e_p \rceil$, where $g_p$ is the exponent of any prime $\mathfrak{p}\mid p$ in the different of $L_\mathfrak{P}/\mathbb{Q}_p$, and $e_p$ is the ramification degree at $p$. The exponent $g_p$ itself can be calculated in terms of higher ramification degrees of $L_\mathfrak{P}/\mathbb{Q}_p$. These facts can be found in the same chapter quoted above, see especially Prop. 4 and display (10) in that chapter.