The following comes from [Birational geometry of algebraic varieties] by Kollar and Mori:
Here are my questions:
1.(The first red line): Why can they assume $Y$ is smooth at the generic point of $Z_0$?
2.(The first and the second red arrows): Does $Z_0$ and $Z_1$ need to be smooth to derive the log discrpancies? In my understanding, in other to compute the log discrepancies, one might need the following blow-up formula([Hartshorne, II, EX.8.5]):
The exercise need the subvariety to be smooth. But I'm not sure if $Z_0$ and $Z_1$ in the proof are smooth subvarieties. For the first red arrow, I guess it has something to do with the assumption that $Y$ is smooth at the generic point of $Z_0$, but I don't know how does the assumption implies $Z_0$ is smooth.
In the setup, $E$ is a prime divisor on $Y$. Hence $\newcommand{\sm}{\textrm{Sm}}$$Y^{\sm} \cap E \subseteq E$ is a nonempty open subset, as is the set $U \subset E$ defined to be the complement of all the components mentioned before the first underline.
Then, $W = (Y^\sm \cap E) \cap U \subset E$ is open, so if we choose $y \in W$, and a codimension one irreducible subvariety $Z_0 \subset E$ containing $y$, it follows that $W \cap Z_0 \subseteq Z_0$ is a dense open subset and hence, the generic point of $Z_0$ is contained in $W \subset Y^\sm$.
As for the next question, $a(E_1, Y, \Delta_Y) = \textrm{ceoff}_{E_1}(D)$ where $$D = K_{Y_1} - g_1^*(K_Y + \Delta_Y).$$
To compute this coefficient, we can restrict to any open subset of $Y_1$ intersecting $E_1$, so in particular we can restrict to $g_1^{-1}(W \cap Z_0^{\sm})$, where $g_1$ restricts to the blowup of a smooth variety along a smooth subvariety, so that Hartshorne exercise applies.
When we restrict to this open subset, we have that $\textrm{coeff}_{E_1}(K_{Y_1} - g_1^*K_Y) = 1$ and $$\textrm{coeff}_{E_1}(-g_1^*\Delta_Y) = \textrm{coeff}_E(-\Delta_Y) = a(E, X, \Delta) = -c-1. $$ The first equality is because $Z_0$ is only contained in the component $E$ of $\Delta_Y$, so the copies of $E_1$ in $g_1^{*}\Delta_Y$ can only come from $g_1^*E = E_1 + ({g_1})_*^{-1}E$.
Then, the same kind of argument works for computing $a(E_2, Y, \Delta_Y)$.