I'm just going to make a quick additional comment on #2 (just to add on to Timo Keller's explanations and Sasha's counter example).

**Theorem (see Matsumura, page 179)** *If $R \subseteq S$ is a finite extension of Noetherian rings with $R$ regular and $S$ Cohen-Macaulay, then $S$ is a flat $R$ module.*

So you just need $Y$ to be regular and $X$ to be Cohen-Macaulay (Matsumura actually has a more general statement which also works in the non-finite case).

On the other hand, if $Y$ is *not* regular then $X/Y$ is very rarely flat. Indeed, since the map is finite, flat is the same as $f_* O_Y$ being a locally free $O_X$-module and we can ask how many (local) $O_X$-summands $f_* O_Y$ has as an $O_X$-module.

**Fact** If $R \subseteq S$ is a finite extension of normal local rings with the same residue field which is etale in codimension 1 (ie, the sort of group quotient that Sasha wrote above), then $S$ has at most one $R$-summand as an $R$-module (in particular, it is nowhere near flat). See the question I asked earlier: Number of free summands of finite local extensions