I allowed Arvel the Swift to live after the first encounter in the spider's chamber and now he has run off. I'm following the quest indicator to catch up with him, but it's getting stuck on a particular doorway, pointing at it even after I've gone through it.
How can I find Arvel and retrieve the Golden Dragon Claw?
I personally never let Arvel escape and always quickly kill him, but I found this information here which might help you out:
Once you free him it becomes clear that he has no intention of giving the claw over to you. He will instead run towards the next room and brush you aside.
Once he reaches the next room full of draugr, he will defend himself, but usually dies. If you save him from the undead, he will be killed by the spike wall trap or the swinging blades trap. If you get too close, he will also turn hostile. Regardless, it is impossible to keep him alive.[verification needed — see talk page] You may also kill him as soon as you free him, although low-level players may find him a useful distraction for the draugr in the nearby room. Arvel carries a journal describing how the claw came into his possession.
Here it says keeping him alive needs to be verified, but if I were you I would double check to make sure he wasn't killed by the draugrs or the spike wall. You're probably not finding him because he died somewhere in the dungeon and you might have missed him.
Let me know if this helped you, otherwise I'll try to find another solution for you.
If the quest marker points to a door, no matter which side of it you're on, this usually means the target is in a completely different cell (ie, outside of Bleak Falls Barrow). What's happening is that the game knows the target is not in your current cell so it points towards a door that will lead you towards the cell it is in - this isn't always the fastest route but can lead through other cells instead of straight to the one where the target is.
I would hazard a guess that since the game is behaving like Arvel is in neither the Barrow nor the Sanctum that he has somehow gotten outside. This shouldn't happen so it is most likely a bug. Plus, if he was outside then the marker should point towards the entrance and exit of the barrow and sanctum, respectively, but the game is famous for not being able to do this very well.
You should be able to verify this by back-tracking where you came from and head back outside via the entrance you came in through. Then, check where the map marker is pointing. If it's back at Bleak Falls Barrow then, as IllusiveBrian said, he may have fallen through the floor where the door to the sanctum is. If not then he is elsewhere in the world; track him down, kill him, take the claw and never trust a Dunmer again.
As noted in comments, make sure this is the only quest active in your journal. If you have verified that, Arvel may have fallen through the floor or otherwise become unreachable, on PC you can try opening the console (~ by default) and use the two commands to attempt to move Arvel (dead or alive) to your location:
prid 00039655
moveto player
If that does not work, you can simply use the following command to add a Golden Claw to your inventory:
player.additem 00039647 1
