Question -- Did the Confederacy have an official procedure for shipping officers and enlisted men burial?
Answer -- None whatsoever.

Question -- Did Confederate government pay costs of transporting bodies home for burial?
Answer -- Absolutely not, in 1861 or later.

Question -- Did families receive official notice of the death, or did that occur if someone (relative, friend) thought to write the deceased's family?
Answer -- The latter - no official notice made. Casualty lists were published in newspapers, but these were unofficial and sometimes inaccurate.

Bear in mind, if Confederates were NOT in control of the battlefield after the fighting ended (Gettysburg, for example), possibility of the dead being given individual or marked burials was extremely limited.