Relation between DVR's of a local domain and localizations of its integral closure. $\textbf{1.}\,\,\,\,\,\,\,\,$
Let $(A,\mathfrak m_A)$ be a one dimensional local domain and let $B$ be its integral closure in the fraction field $L=\textrm{Frac}\,A$. Assume that $B$ is finitely generated over $A$.
I would like to understand why the discrete valuations rings of $L$ dominating $A$ $are$ the localizations $B_{\mathfrak m}$ of $B$ at its maximal ideals.
I was able to show that such a localization is a DVR of $L$. But, just to conclude this first step, how does one prove that $\mathfrak m_A\subset \mathfrak mB_{\mathfrak m}$?
We have
\begin{equation}
A\hookrightarrow B\hookrightarrow B_{\mathfrak m} \,\,\,\,\,\,\,\,\,x\mapsto x/1
\end{equation}
but how do we know that $x\in\mathfrak m_A$ goes to something in $\mathfrak m$ via the first map?
And, for showing the "converse" (from DVR to localization) I have no ideas.
$\textbf{2.}\,\,\,\,\,\,\,\,$
Afterwards, I would like to understand the following: suppose we have an integral variety $X$ with normalization $\pi:\tilde X \to X$. Let us take a (closed) codimension one subvariety $V\subset X$. The questions is: why do we have a correspondence between the DVR's of $L=K(X)$ dominating $A=\mathscr O_{X,V}$ and the (closed) subvarieties $Z\subset\tilde X$ mapping onto $V$? I feel like domination should translate $\pi(Z)=V$, but can't see it neatly.
Thank you!
 A: (1).1. Let $P=\mathfrak mB_\mathfrak m\cap A$. Then $A/P\to B_\mathfrak m/\mathfrak mB_\mathfrak m$ is injective and integral. As the RHS is a field, this implies that $A/P$ is a field, hence $P=\mathfrak m_A$ and $\mathfrak m_A\subseteq \mathfrak mB_\mathfrak m$.
(1).2. Let $R$ be a DVR of $L$ dominating $A$. Any element $b\in B$ is integral over $R$ and belongs to $\mathrm{Frac}(R)$, so $b\in R$ because $R$ is integrally closed. Hence $B\subseteq R$. Let $Q=\mathfrak m_R\cap B$. Then $Q$ is a prime ideal of $B$ containing $\mathfrak m$, hence non-zero. As $\dim B=1$, $Q$ is maximal and $B_Q$ is a DVR of $L$ dominated by $R$, hence $B_Q=R$. 
(2). One can apply (1) to $O_{X,V}$. The subvariety $Z$ gives rize to a DVR equal to $O_{\tilde{X}, Z}$. As $Z$ surjects to $V$, its generic point goes to the generic point of $V$, hence $O_{X,V}\to O_{\tilde{X}, Z}$ is a homomorphism of local rings. Conversely, a DVR dominating $O_{X,V}$ gives rize to a point $\xi$ of $\tilde{X}$ lying over the generic point of $V$ by 1.2. Let $Z$ be the Zariski closure of $\xi$. As $\pi$ is a finite map hence closed, $\pi(Z)$ is closed and dense in $V$, hence equals to $V$. 
