Let $k$ be a field, $S=k[x_1,\dots,x_r]$ and $M$ a finitely-generated, graded $S$-module.
Definition: We say that $M$ is weakly $m$-regular if $Ext^j(M,S)_n=0$ for all $j$ and $n=-m-j-1$. We say that $M$ is $m$-regular if $Ext^j(M,S)_n=0$ for all $j$ and $n \le-m-j-1$.
Theorem 20.17 in Eisenbud (CA with a view...): With the notation above, let $N$ be the maximal submodule of $M$ having finite length. If $M$ is weakly $m$-regular, then $M/N$ is $m$-regular.
The proof of the theorem proceeds by induction on $\dim M$. If $\dim M>0$, Eisenbud considers a short exact sequence $0 \rightarrow N \rightarrow M \rightarrow M/N \rightarrow 0$ and the corresponding long exact sequence of $Ext(-,S)$.
Question 1: Why is it true that $Ext^j(N,S)=0, \forall j<r$? As i understand, $r$ is the number of indeterminates, what does this have to do with $Ext^j(N,S)$? Eisenbud invokes proposition 18.4, but this proposition applied to $Ext^j(N,S)$ simply says that $Ext^j(N,S)=0$ for any $j$ less than the grade of $N$, which is a definition.
Question 2: Even if $Ext^j(M/N,S) \cong Ext^j(M,S), \forall j<r$, why does the weak $m$-regularity of $M$ imply weak $m$-regularity of $M/N$? How about $j \ge r$?
Question 3: Why is $(x_1,\dots,x_r)$ not inside any associated prime of $M/N$?