divisor on curves

divisor on curves

On proof of Hartshone's book Proposition (II.6.9). I have some question
Proposition (II.6.9) is that
$f:X \rightarrow Y$ be a finite morphism of nonsingular curves. Then for
any divisor $D$ on $Y$,
$\operatorname{deg}f^*D=\operatorname{deg}f\operatorname{deg}D$.
In proof, let $Q$ be a closed point of $Y$ and let
$V=\operatorname{Spec}B$ be an affine open of $Y$ containing $Q$. Let
$K(X),K(Y)$ are the function field of$X,Y$, respectively. Note that if $A$
is the integral closure of $B$ in $K(X)$, then
$f^{-1}(V)=U=\operatorname{Spec}A$. Let $m_Q$ be the maximal ideal of $B$
and $t$ is a local parameter at $Q$ and let $C=A_{m_Q}$. Let $P_i \in X$
be such that $f(P_i)=Q$ and $m_i$ be a maximal ideal of $C$ corresponding
to $P_i$.
I have three questions:
if $[K(X):K(Y)]=r$, then why $\operatorname{dim}_k(C/tC)=r$?
$tC=\cap_i(C_{m_i}\cap C)$?
$\operatorname{dim}_k(\mathcal{O}_{P_i}/t\mathcal{O}_{P_i})=v_{P_i}(t)$?
where $v_{P_i}$ is a valuation.