Can you kill the Drake in the Undead Burg

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.

You are correct in saying that stat scaling doesn't really improve at all, beyond a minor improvement from ascensions (a sword might go from E --> D for example). In general, the stat classes that each weapon has in the game won't change much, even after being ascended fully.

Compared to Demon's Souls, upgrading a weapon to its max in Dark Souls provides less benefit because of this.

The bonus damage from stat scaling does increase for +n weapons, albeit it's a minor, linear increase (for example, a Battle Axe from +0 to +2 made its bonus damage from stats go +20->+22->+24). Thanks Tacroy

If you need to know when to replace your Drake's Sword, I would say get rid of it when you have high enough stats in one of the "damage" stats (Strength/Dexterity/Faith) and you have an adequate amount of resources to reinforce a scaling weapon to its max.