This problem is from Serre's Local Fields book. I am having difficulty trying to understand what he means by "sufficiently near $x$".
Suppose that $B$ (hence also $A$) is a discrete valuation ring. If $\bar{L}$ and $\bar{K}$ denote the residue fields of these two rings, suppose also that the extension $\bar{L}/\bar{K}$ is separable. Then if $B=A[x]$ and $y$ is sufficiently near $x$, then $B=A[y]$.
Concerning the first question: every discrete valuation ring carries with it a natural distance $d$ induced by the valuation $\nu$: $d(x,y)=2^{-\nu(x-y)}$. So, $y$ is sufficiently close to $x$ is $\nu(x-y)$ is large enough.