They definitely had to come up with a body for him unless they went with a body-snatch or clone. Anything's plausible in comics!
The Bamfs are shapeshifters by nature, and they seem to take on the qualities of whoever feeds them. By that logic, they initially gained teleportation through Azazel's blood. Would they have gained it through being fed bits of Kurt's soul? It's hard to say because he didn't get to them first. Given that Kurt used Azazel's blood magic against him, I'm guessing that Kurt's blood is not his own. Does that mean he picks up Daddy's longevity? It'd be impossible to tell if any shapeshifting capabilities come from the Bamfs or Azazel, and the latter was shown shooting paralyzing energy during his first appearance. Was that sorcery or a natural ability? Kurt could have inherited quite the power set.
I think you're right that Kurt will probably still consider himself a mutant. Like you said, though, a virus would reveal the truth of the matter. A DNA test presumably would as well. Will he be more susceptible to magical attacks? And what exactly does death mean to him now?