Question 1: The main reference on algebraic stacks (Laumon and Moret--Bailly) defines a separable algebraic stack as one having universally closed diagonal. For schemes separability is simply defined by the condition that the diagonal is a closed immersion. Why this difference?

Question 2: Presentations of algebraic stacks are defined (in LMB) as morphisms (with some properties) $X\to\mathscr{X}$ with $X$ an algebraic space. What does one lose by only considering schemes $X$ instead?

These questions are certainly well-known to experts in stack theory and I know there are a few around on MO, so I'm hopeful that I will be sufficiently enlightened.