Are targets chosen for Cull the Weak when it is put on the stack or when it resolves?

No — Cull the Weak never targets. Its text "Each player kills one of their units" falls under Rule 355.10.e, so each player independently chooses a unit as the spell resolves, not when it's put on the chain. Nothing is locked in at play, and you can even cast it with an empty board — if you have no units at resolution, you just don't kill one (do as much as you can). Because it doesn't target, it also bypasses Deflect and cards like Not So Fast that require targeting.