I know if you cut off the Drake’s tail you get the Drake Sword, but is it also possible to kill him? I know in Demon’s Souls there were a couple dragons which couldn’t be killed when you first met them, and would simply fly away after their health was low enough.
Does the same thing happen here, or is it possible for me to kill the Drake? If I do kill him, what is the reward?
Yes you can indeed kill him, and you get 10 000 souls when you do. As far as being killed with arrows, unfortunately I could not get a clear answer. The consensus on the forums I’ve looked through seems to be that he’ll get down to about half health, at which point he’ll cover his head, seemingly healing himself. There have also been several glitches reported where he died randomly.
Here is a video of somebody accomplishing the task: