Aye, Yamato is first and foremost a sword of Sparda that he passed on to Vergil, the same with Rebellion to Dante. Infact, look carefully at it whenever in the cutscenes. Yamato is always surrounded in a purple Aura when Nero holds it, not blue.
You actually can see a LOT of InuYasha parallels in the Twins; the older coldhearted brother who hates humans, and the younger hotheaded brother who finds himself defending them., as well as the blades they inherit from their great legendary father.
As such, Yamato certainly reacted to the blood of Sparda's lineage in Nero, though the fact his own presence and awakening ALONE repaired and healed it says there's a much stronger connection than merely being descended. The idea that Nero is Vergil's son is in fact contradictory to the statements of the developers that Nero was not directly related to the twins. Plus, you have to retcon entire timespans and ages extensively to make Nero Vergil's bastard son. For one, Vergil was 19 in DMC, which is ten years before DMC4. That means even if Nero were 17, Vergil would have to be 11 or 12 when he got some woman pregnant. You also have to understand, a 19 year old, from the PoV of someone Kobiyashi's age, IS a child.
That and as pointed out, the idea of Vergil randomly sleeping with a human woman, let alone a prostitute, is so demeaning to his character that it's basically Bingo's fanwank and whining on his ideas not being used. No doubt, Bingo wanted to make Nero Vergil's son, and was overruled. With how much Deadly Fortune rewrites and contradicts important parts of the game, it really is just his method of trying to force his ideas as "fact". I have little doubt the heads of the project said "fine, write a book, but it doesn't really count." Why novelize the game, when people play it anyway and notice the problems in comparing the two?
On the Tony matter. Episode 8 of the anime explicitly has Dante recall having gone by the name of Tony when he was younger, a clear reference to the events of the first novel...as if the strawberry sundaes weren't enough of a reference. And the Tony in question did in fact look just like a young Dante, and the history sounded suspiciously like Dante's own past; on the run with a mother, hunted by demons. Sounds like it it really was Dante, and he had blocked out the memories, since everyone seemed to recognize him in that episode on sight as Tony, then the mother in question would have actually been Nell Goldstein raising him, confirming the theory that she was his "adopted" mother after Eva died.