What happened to the corruption? You need to be a bit more specific with that question.

[ As for the other one: ]

The unique cold of Dragonspine Mountain has frozen his blood and prevent the corruption from spreading. In its current state, the stones are safe to leave unattended. Though I wouldn't recommend you keep one with you. Vestiges of the dragon's will still remain inside the blood stones. If you were to unfreeze it, though, anyone or anything that comes into contact with the blood will fall to destructive urges.