I have a quest for the blades to kill a dragon at Autumnwatch Tower.
I also have a quest from someone else to kill the dragon at Dragontooth Crater.

Both of these locations have been cleared prior to me getting the respective quests, so their dragons are gone. There was a dragon at the crater when I fast-traveled to it, but it wasn't interested in me and flew off after I attempted to get its attention with my arrows.

There is no dragon at the tower, the quest marker is hovering in the air over nothing.

I am playing on the PC, so I tried advancing the quest stages. After searching on UESP, I found that these both share the same quest ID (BQ04). I set the stage to 200 (completed) and I received some gold, but both still remain in my log as not finished, and I don't think I'll be able to hunt any more dragons with the Blades since I'm technically still on a mission for them. What do?

Try using the console command to make them complete instead of just getting the gold part. If that doesn't work try going back a few saves maybe?

I have those same problems as well; however, I've yet to go there and see if there are dragons flying about. I did not get them added to my Quest Log though. I actually got these from City Guards telling me they see dragons when I walked by them, are these City Guards the "someone else" you are talking aboutz?

Had a similar problem - chasing a bounty on a dragon only to find a hovering arrow over nowhere. In this case I think the dragon was spawned by the quest but then drifted to the nearby giant camp and got stomped before I exited fast-travel. That's my theory anyway.

Had a companion rescue mission to Mara's eye pond once - fast travelled to them to find that the kidnapped unfortunate had already walked half-way home (presumably not caged properly in Mara's eye pond). They had reached their home town by the time I caught up with them, but they were still ever so grateful for being rescued and begged for an escort home. Quest related spawns do move even when you're a long way away from them.