What was the general breakdown of casualties per their cause during American Civil War?
E.g. bullets vs. artillery vs. edged weapons vs. disease vs. natural death.
Did that breakdown change in meaningful way between the beginning and the end of the war?
You can get a breakdown of the major causes of death here.
Prior to the 20th century (possibly late 19th), the dominant cause of death in war was disease: the troops were in close quarters with unsanitary conditions and inadequate means of handling these. This number was followed by complications related to injuries actually suffered in battle — frequently one would get a small injury and it would get infected and kill the individual even though it would be trivial to heal the wound in better conditions.
Unfortunately, I cannot give you a specific breakdown of blade vs. bullet (and I suspect that it would be impossible to do so), but I will say that, based on the artillery technology of the time, it is fairly safe to say that discerning the difference between bullet and canon shot would be quite impossible. Even if you assume that there was a substantial interest in maintaining those records, there would not have been effort made to determine which injuries are post-mortem.
Bruce Catton's history and Shelby Foote's history should have the details. I cannot check as my copies are at home. From what I remember, it was pre-battle medical care and unsanitary conditions that killed the most soldiers. The first use of modern rifles with Napoleon tactics contributed to the hight number of dead on the battle field.