It is possible to get all the pieces and repair the key without getting the updates for investigating the locations which means you have the repaired key but don't have the achievements.
Yep, this happened to one of my characters before I realized what was going on. I had to drop the quest (I also destroyed the key for good measure) and do it again. I did learn however that if you have the key pieces but don't have the location update, you can go there and right-click the appropriate fish to get the update. This assumes you haven't repaired the key at that point. If you made the key already I'm pretty sure the only option is to drop and re-quest.
Yep.I repaired the key because I got all the parts from killing the bears. If it's meant to be a solo mission then the whole group shouldn't get the key parts.
Update: I destroyed the key and redid the entire quest. The last step should recognize if you already have the 'repaired' key in your inventory and update..