Confusion about Vakil's proposition 17.4.5, on finite morphisms between curves

I've now understood what was confusing me, and will post this answer for future readers who may have the same problem:

A regular scheme is, by definition, locally Noetherian! The confusion arose from the fact that we only define regular points for locally Noetherian schemes, and so $C'$ is in fact locally Noetherian, which I didn't realise at the time.

Given this, I can answer my questions as follows:

Firstly, at the same time as assuming $t \in A'$, we may assume that $\mathfrak{m} = (t)$, since $\mathfrak{m}$ has finitely many generators, and considering each as an element of $A'_{\mathfrak{m}}$ we see that each is of the form $t \cdot f/g$ for some $f/g \in A'_{\mathfrak{m}}$ and so by considering the distinguished affine open piece corresponding to the product of all such $g$ (of which there are finitely many) we have that $(t) = \mathfrak{m}$. It is now clear that any prime of $A$ containing the image of $t$ pulls back to a prime ideal containing $(t)$, and so equaling $\mathfrak{m}$ by maximality.

Secondly, any non-empty fibre of an integral morphism has dimension $0$ (exercise $11.1.E$ in Vakil) so, under finite (even integral) morphisms, only closed points can map to closed points (if a non-closed point mapped to a closed point, so would everything in it's closure by continuity and then the fibre would have dimension strictly positive).