A couple of them do not count though: http://www.uesp.net/wiki/Skyrim:Oblivion_Walker#Achievements
Also I have read that if you use the console then that takes away the achievement (I suppose to stop people from using console commands to just clear all daedric quests). I don't know if this is true or not though.
True.
The Mask of Clavicus Vile is an Artifact, but the Axe is not. So to get the Achievement you have to turn in the Axe for the Mask.
With Hircine, the Ring is the Artifact, though the Savior's Hide also works from what I've heard (I got the Ring in my game, and I have the achievement unlocked).
With Nocturnal, the Skeleton Key is iffy at best. I've heard of people unlocking it as soon as they grabbed the Skeleton Key, but I've also had others (myself included) who have carried the Skeleton Key around for months, and it never worked (so I returned it and then went and did another Daedric Quest...)
With Azura, either of the Stars will work (Azura's Star or the Black Star). It's been proven that getting either of these will work towards the Achievement/Trophy.