Cobalt:Nowhere in the game did they say he was a descendant of Sparda. At least not to my recollection anyways. Everyone said he had "The blood of Sparda." So in this case it could be from a number of things. I think he obtained the arm itself from the corpse of Nelo Angelo. Along with Vergil's soul. And when he came in contact with Yamato he obtained the DT. So in other words, the power, DT, AND the blood came from Vergil's arm.
Very good point, Cobalt. I really like your theory.
And yes, Nero is not Sparda's descendant. He just inherited his power like Sanctus said. Nero wasn't able to go into Devil Trigger until he got Yamato. He needed it to have enough strength to save Kyrie.