What is the Shar'i (Islamic legal) classification of Neanderthals? Did they exist before Adam or after him? What is their nature if their form differed from humans? Are they held accountable (mukallaf)?

Adam is the father of humanity, and there was no humankind before him. If the creature mentioned in the question is a human, then it must be from the progeny of Adam and accountable, for Allah Almighty says: "And I did not create the jinn and mankind except to worship Me." Delving into such topics must be based on knowledge and clear evidence, not merely on conjecture and speculation.