For DuVal singularities, $K^2$ and $\chi({\cal O}_X)$ are the same as the ones of the minimal desingularization. In many cases, it is more convenient to fix a minimal desingularization and do the computations there. But then, if $Y\to X$ is such a minimal desingularization of a surface with DuVal singularities then $b_2(X)<b_2(Y)$ (since $(-2)$-classes get collapsed), whereas (I think) $b_1$ stays the same. As I said $\chi$ and $K^2$ of $X$ and $Y$ coincide, so Noether's formula does not hold for $Y$ without modifications.

Did you already have a look at Chapter VIII of "Compact Complex Surfaces" by Barth, Hulek, Peters, van de Ven; as well as S. Kovacs' paper "The cone of curves of a K3 surface"?

just a comment: if $T$ has a non-trivial invertible sheaf ${\cal L}\in{\rm Pic}(T)$ then the projectivization of ${\cal L}\oplus{\cal O}_T$ gives a non-trivial $\mathbb{P}^1$-bundle over $T$. For $T=\mathbb{P}^1$ we get the Hirzebruch surfaces Francesco already mentioned -- but it also produces non-trivial families of $\mathbb{P}^1$'s over affine (in particular: non-proper) bases. In particular, this highlights even further that $D_\epsilon$ is the only reasonable choice.

sorry, of course you're right: if the ramification is not tame then (by the same arguments as in the $1$-dimensional case), the contribution of $R_i$ is strictly larger than $e_i-1$.