> Is Alex/Alexander a nickname for Sasha

it's the other way around: "Alexandr" is the "canonical" name, and "Sasha" is a shorter version of it

In Russian some names can be used with "sha" suffix that makes the name sound "softer": in many cases it's how parents call their kids.

Mikhail -> Misha; Mariya -> Masha; Alexandr(Male) or Alexandra(Female) -> Alexasha -> Sasha