The obvious 'fluff' quotation here is:
[quote author=Originally published in WD127:
When Kaela Mensha Khaine, the Bloody Handed God of the Eldar, fought with Slaanesh the Lord of Pleasure, he was quickly overwhelmed and his energy captured by the newborn God. For the Bloody Handed God was as much a part of Slaanesh as of Khorne - being a product of that part of the Eldar nature which finds gratification in murder and pleasure in bloody violence. Khorne the Blood God, the Patron of War, Murder and Battle, roared with rage to discover one of his own taken from him in this way. Then Khorne and Slaanesh clashed headlong, the Blood God fighting to recover the portion of his power that had been robbed from him, Slaanesh driven by his uncontrollable hunger to consume everything in his path. The Bloody Handed God of the Eldar was tossed this way and that, at first grasped by Slaanesh, then tugged back into the compass of Khorne.
Eventually the rage of the Blood God and the passion of the Lord of Pleasure were exhausted, and the boundaries between them were established. Like a leaf in the eye of a hurricane, Kaela Mensha Khaine fell among the calm, down through the Realm of Chaos and into the material universe. As he entered the material universe he divided into many shards of energy, scattering his power so that neither Khorne or Slaanesh could ever find him again. Each shard entered the body of an Eldar, filling the body with his own mind, possessing it, so that it became a virtually indestructible blood-lusting murderer - the material manifestation of the Bloody Handed God. These are the Avatars of the Bloody Handed God.
Not a great deal to add at this point since, for me, the answers to the questions are a quite simple:
1) No.
2) Because one is a Chaos God and the other a Warp Entity bound by the conceptions of a race of the matterium (i.e. the eldar).
3) Yes.
Kage