What are some benefits of fixing a lazy eye?

Cosmesis. Often in teenagers or adults requests are made to straighten a strabismic eye. This is done primarily for psychsocial reasons to make the eyes appear more natural. Frequently, the eye misaligment is worse in one eye and surgery is often performed on the amblyopic eye only. The vision however will not improve.
Crossed eyes. That is two questions. In childhood a lazy eye needs to be treated so that vision can improve in that eye. Also, surgery in childhood not only helps appearance but can also help stereovision to develop of the eyes are aligned by surgery. Eye muscle surgery in adults is generally done to restore normal appearance.It is not cosmetic in that normal anatomy is being restored.This is important note.