Five days after a corneal abrasion, my eye feels better. When will my sight return?

Variable. Depends on size, depth, location, healing rate, tear film. Usually an epithelial abrasion does not result in a scar. So once healed, vision returns.
Usually, yes! In most cases, corneal abrasions completely heal leading to normal vision. We most often are concerned about recurrent corneal erosion or where the corneal surface re-opens spontaneously at the site of injury. This is best prevented with lots of artificial tears.