Wolverine did regrow his claws, it was because of his regeneration powers.
Cyber was killed by someone named Genesis. Genesis then tried to recreate the bonding process used on Logan in the Weapon X program with Cyber's skin, it didn't work, Logan's body rejected the adamantium. Wolverine didn't get his adamantium back until Apocalypse turned him into the new Horseman of Death (I think Wolverine was the 4th Horsemen of Death, I do remember Angel was the first).