I would like to understand this notion better; where could I find some examples? In particular, I am interested in the following questions (and references for the answers).

Is a universal homeomorphism of connected regular (excellent finite dimensional) schemes an isomorphism if these schemes are not positive characterstic ones?

Suppose that a finite morphism $f:X\to Y$ of connected regular (excellent finite dimensional) schemes is generically purely inseparable. Does $f$ restrict to a universal homeomorphism of some open (non-empty) subschemes of $X$ and $Y$?