The Bridge Wyvern is the official name of this dragon. As far as I know, it will never re-locate and will impede your passage on one end of the bridge or the other.
On the first playthrough, the Wyvern has 2208 HP. It would take you a while to pelt him down with arrows, but it is viable. If you're only doing 15 damage, it might be a good idea to check you meet the minimum stat requirements of the bow you're using as there's a damage and attack speed penalty if your parameters are lower.
That being said, the Bridge Wyvern is highly prone to glitching, and dying as a result. As mentioned on the Dark Souls Wiki, here are some methods of causing him to glitch and die, as well as how to safely kill him with arrows:
Ways to kill the Hellkite Wyvern without endangering your character:
- Shoot the wyvern from below the bridge with an arrow and use heavy
soul arrow when it lands on the bridge and turns around to cause a
glitch death. (may have been/should be patched soon)
- Shoot the wyvern from above and behind it after you gain access to the Black Knight's bridge tower (located above 3 plague rats) to
cause a glitch death. (Hellkite Wyvern may or may not appear due to
glitch. Exit area/zone and re-enter. It may be necessary to go as far
as the Darkroot Garden to get it to come back)
- Shoot the wyvern from under the bridge; first cutting off the tail then shooting the head/neck. At base stats needed for the Short
Bow (purchased for 1,000 souls from Undead Merchant (Male)) it will
take roughly 350-400 arrows to take him down.
There are numerous other ways that the dragon could glitch and die. I have also caused it to glitch during soul farming - I would repeatedly trigger his fire attack to kill the enemies on the bridge and go back to the nearest bonfire to reset the monsters.
In terms of rewards - You receive 10,000 souls for his death. If you keep shooting his tail, it will eventually be cut off and you'll be rewarded with the Drake Sword, a 200 damage base sword that is good for starters. I suggest doing this before killing him.
I believe these are the resonance areas for the Way of White covenant. If you cast a miracle there, it will have increased effectiveness under certain circumstances. It's tied into the online/multiplayer aspect of Dark Souls, and I'm not sure it's been fully explored and documented.
Best Answer
Originally it was fairly difficult to obtain arrows easily, as the Male Undead Merchant in the Undead Burg and the Giant Blacksmith in Anor Londo were originally the only two merchants to sell any kind of arrows, but now almost every merchant and certainly every blacksmith sells arrows and bolts, so as GnomeSlice suggested, farm souls and then buy the arrows from a merchant.