I already covered that. Vergil probably killed Arkham because he had become a demon and was obviously up to no good. I'd say Vergil realized that Arkham was dangerous when he gripped the Yamato and went down the blade. Now, either Vergil intended for Arkham to receive his Devil Trigger and become Jester (so that they could get it over with), or he actually wanted to kill him. But since all demons get a Devil Trigger, I'd assume Vergil knew he would live. Call it Vergil's way of punching him in the face to knock him out cold.
Why would it matter that he punches the head of a corpse? It's literally dead meat. Maybe he lacks respect for Beowulf, but that doesn't mean Vergil is completely heartless.
And Beowulf was dangerous. Beowulf was made to kill. He intended to kill Sparda's sons, so I think that justifies killing him. It's self-defense... at some point, Beowulf would've come after Vergil, and he would have to kill him. Why waste time? Why not kill him right there? I think that's Vergil's reasoning.
Like I said, I didn't mean Vergil was honorable per se, just that he didn't kill humans or demons that didn't mean any harm. He could've killed Lady or Arkham at any point. Heck, he could've even killed Dante before he got so far up the tower. So he's not necessarily honorable, just... not insanely immoral like new Vergil.