Does Unyielding Spirit stop Cull the Weak?

No, Unyielding Spirit does not stop Cull the Weak.

According to the official RiftJudge FAQ (ID 3860): "Ruling: No, Cull does not prevent Unyielding because Cull is not damage."

Unyielding Spirit only protects a unit from damage dealt by spells and abilities. Cull the Weak is a "kill" effect, not a damage effect, and therefore bypasses the protection provided by Unyielding Spirit.